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ICML/2025/Learning Safe Control via On-the-Fly Bandit Exploration/full.md

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+# Learning Safe Control via On-the-Fly Bandit Exploration
+
+Alexandre Capone $^{1}$  Ryan Cosner $^{2}$  Aaron Ames $^{2}$  Sandra Hirche $^{3}$
+
+# Abstract
+
+Control tasks with safety requirements under high levels of model uncertainty are increasingly common. Machine learning techniques are frequently used to address such tasks, typically by leveraging model error bounds to specify robust constraint-based safety filters. However, if the learned model uncertainty is very high, the corresponding filters are potentially invalid, meaning no control input satisfies the constraints imposed by the safety filter. While most works address this issue by assuming some form of safe backup controller, ours tackle it by collecting additional data on the fly using a Gaussian process bandit-type algorithm. We combine a control barrier function with a learned model to specify a robust certificate that ensures safety if feasible. Whenever infeasibility occurs, we leverage the control barrier function to guide exploration, ensuring the collected data contributes toward the closed-loop system safety. By combining a safety filter with exploration in this manner, our method provably achieves safety in a setting that allows for a zero-mean prior dynamics model, without requiring a backup controller. To the best of our knowledge, it is the first safe learning-based control method that achieves this.
+
+# 1. Introduction
+
+With the growing proliferation of robotics in safety-critical fields, e.g., autonomous vehicles, medical robotics, and aerospace systems, the need for methods that ensure the safety of systems and their users has become paramount. In control tasks, guaranteeing safety has become synonym-
+
+*Equal contribution  $^{1}$ Robotics Institute, Carnegie Mellon University, Pittsburgh, PA, USA  $^{2}$ Department of Mechanical and Civil Engineering, California Institute of Technology, Pasadena, CA, USA  $^{3}$ TUM School of Computation, Information and Technology, Technical University of Munich, Munich, Germany. Correspondence to: Alexandre Capone <acapone2@andrew.cmu.edu>.
+
+Proceedings of the  $42^{nd}$  International Conference on Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).
+
+mous with guaranteeing that the system always satisfies a pre-specified set of constraints. While there exist various tools to achieve safety whenever the system is known perfectly (Agrawal & Sreenath, 2017; Ames et al., 2017; Bansal et al., 2017; Wabersich & Zeilinger, 2018; Zeng et al., 2021), doing so if the dynamics have to be learned poses a significantly more difficult challenge.
+
+To address safety in a control setting, existing approaches typically attempt to bound the model error and assess how its worst-case estimate affects safety. To estimate the model error, frequently used frameworks include ensembles of neural networks (Chua et al., 2018; Curi et al., 2020) or Gaussian processes (GPs) (Capone et al., 2022; Rodriguez et al., 2021). The impact of the model error on safety is then frequently estimated by predicting the worst-case trajectory multiple steps into the future (Akametalu et al., 2014; Hewing et al., 2020; Koller et al., 2018). Alternatively, safety filters attempt to condense the long-term impact of a control input on safety (Ao et al., 2025; Berkenkamp et al., 2017; Castaneda et al., 2021; Cheng et al., 2019; Jagtap et al., 2020). A crucial drawback of most of these methods is that they assume the model uncertainty is always small enough for the conservative safety constraints to be feasible. In other words, if the uncertainty becomes too large, the underlying methods are no longer applicable.
+
+Motivated by the aforementioned challenges, we present a learning-based control algorithm that ensures safety by exploring on the fly. We use a Gaussian process model to estimate model uncertainty, which we combine with a control barrier function to compute a robust safety filter. Whenever our safety filter becomes infeasible, we employ the control barrier function to guide exploration in a manner that is informative for safety. Formally, this is done by leveraging the lower confidence bound of the certificate constraint function for certifiably safe control inputs, and the upper confidence bound for exploration. Our approach draws heavily from ideas from Bayesian optimization, where exploration is done in a similar manner. Overall, our main contributions are as follows:
+
+1. We present the first safe control algorithm that addresses infeasible safety filters by exploring online using a Gaussian process bandit algorithm.
+
+- Starting state - Feasible (safe) region - Infeasible (safe) region - Unsafe region - System trajectory
+
+![](images/b7d90696de64990b74e2ab19bb30dc51c82933d1eff297d6ad8c914c9f299e2c.jpg)  
+(a) System state exits the region where the safety filter is feasible.
+
+![](images/90cd0926c879b18c2cc22ed4bd7f3634c03dfd5f945556dfe8aee92f97ae6787.jpg)  
+(b) Our algorithm computes exploratory control inputs using a bandit-type technique.
+
+![](images/e4be0989ac260f23d679e64ad18028836107ef67e878088c962af55f64b5c7f8.jpg)  
+(c) Theorem 3.2 guarantees that the safety filter always becomes feasible before the boundary of the safe set is reached.  
+Figure 1. Illustration of our method (Algorithm 1) and theoretical result (Theorem 3.2).
+
+2. We rigorously analyze our approach and discuss our main theorem, which shows the interdependence between model uncertainty, sampling time, and safety.  
+3. We showcase our approach using two numerical simulations where we achieve safety using a control barrier function, a dynamics model with a zero-mean prior, and no prior measurements. To the best of our knowledge, this setting cannot be addressed by current state-of-the-art methods.
+
+The remainder of this paper is structured as follows. In Section 2 we describe the problem setting considered in this paper, together with some background on control barrier functions and Gaussian processes. Our main contribution, which includes an exploration-driven safe control algorithm and safety guarantees, is presented in Section 3. In Section 5, we apply our method using numerical simulations of cruise control and a quadcopter. We finalize the paper with some conclusions in Section 7.
+
+# 2. Background and Problem Setting
+
+We consider a system that is affine in the control inputs
+
+$$
+\dot {x} = f (x) + g (x) u \tag {1}
+$$
+
+where  $x \in \mathcal{X} \subseteq \mathbb{R}^n$  and  $u \in \mathcal{U} \subseteq \mathbb{R}^m$ , and  $f: \mathbb{R}^n \to \mathbb{R}^n$  and  $g: \mathbb{R}^n \to \mathbb{R}^{n \times m}$  are (partially) unknown locally Lipschitz continuous functions that represent the drift dynamics and the input matrix, respectively. Eq. (1) represents a broad class of physical systems. Though our method can be straightforwardly extended to the more general case  $\dot{x} = f(x,u)$ , this structure significantly facilitates the computation of safe control inputs, as described in Section 2.1. We further assume that  $\mathcal{X}$  is an open and connected set and that  $\mathcal{U}$  is compact. When a locally Lipschitz controller
+
+$\pi : \mathbb{R}^n \to \mathbb{R}^m$  is used, we can define the closed-loop system:
+
+$$
+\dot {x} = f (x) + g (x) \pi (x). \tag {2}
+$$
+
+We also assume to know an upper bound  $L_{\dot{x}} \in \mathbb{R}_{+}$  for the time-derivative of the dynamics, as specified by the following assumption.
+
+Assumption 2.1. There exists a known positive constant  $L_{\dot{x}}$ , such that  $\| \dot{x} \|_2 \leq L_{\dot{x}}$  holds for all  $x \in \mathcal{X}$  and  $u \in \mathcal{U}$ .
+
+We define safety by considering a safe set  $\mathcal{C}$ , as specified in the following.
+
+Definition 2.2 (Safety). The closed-loop system (2) is said to be safe (with respect to  $\mathcal{C}$ ) if  $x(0) \in \mathcal{C}$  implies  $x(t) \in \mathcal{C}$  for all  $t \geq 0$ .
+
+We consider the case where the safe set  $\mathcal{C}$  corresponds to the superlevel set of some known continuously differentiable function  $h:\mathbb{R}^n\to \mathbb{R}$  with 0 a regular value:
+
+$$
+\mathcal {C} \triangleq \{x \in \mathbb {R} ^ {n} \mid h (x) \geq 0 \}.
+$$
+
+This specification of  $\mathcal{C}$  allows us to achieve safety through the control barrier function framework, which we discuss in the following section. This paper considers a compact  $\mathcal{C}$  and a Lipschitz continuous functions  $h$ , as stated in the following assumption.
+
+Assumption 2.3. The safe set  $\mathcal{C}$  is compact and the function  $h$  admits a Lipschitz constant  $L_{h}$ , i.e.,  $|h(x) - h(x')| \leq L_{h} \|x - x' \|_{2}$  holds for all  $x, x' \in \mathcal{X}$ .
+
+Given an arbitrary nominal controller  $\pi_{\mathrm{nom}}: \mathcal{X} \to \mathcal{U}$ , we aim to design a control law  $\pi$  that renders the closed loop
+
+system (2) safe while following  $\pi_{\mathrm{nom}}$  as closely as possible. We make the following assumption regarding the nominal controller:
+
+Assumption 2.4. The nominal controller  $\pi_{\mathrm{nom}}$  is locally Lipschitz in  $\mathcal{X}$ .
+
+Assumption 2.4 is not restrictive, as it includes a broad class of controllers, including most types of neural networks.
+
+# 2.1. Safety through Control Barrier Functions
+
+Our method attempts to ensure safety using control barrier functions, which we introduce in the following.
+
+Definition 2.5 (Control Barrier Function (CBF) (Ames et al., 2017)). Let  $\mathcal{C} \subset \mathcal{X}$  be the 0-superlevel set of a continuously differentiable function  $h: \mathbb{R}^n \to \mathbb{R}$  with 0 a regular value. Furthermore, let  $\alpha$  be a radially unbounded, strictly monotonically increasing function with  $\alpha(0) = 0$ . The function  $h$  is a control barrier function (CBF) for (1) on  $\mathcal{C}$  if, for all  $x \in \mathcal{X}$ ,
+
+$$
+\sup  _ {u \in \mathcal {U}} \dot {h} (x, u) \triangleq \sup  _ {u \in \mathcal {U}} \frac {\partial h}{\partial x} (x) (f (x) + g (x) u) > - \alpha (h (x)). \tag {3}
+$$
+
+Intuitively, (3) states that there exists a control input that decreases the time derivative of the CBF as the boundary of the safe set  $\mathcal{C}$  is approached, effectively slowing down the system as the boundary gets closer. If  $f$  and  $g$  were known perfectly, we could leverage this property to compute certifiably safe control inputs by including a CBF increase condition as a constraint when synthesizing control inputs:
+
+$$
+\begin{array}{l} \pi (x) = \underset {u \in \mathcal {U}} {\operatorname {a r g m i n}} \quad \| u - \pi_ {\text {n o m}} (x) \| _ {2} \tag {4} \\ \begin{array}{c c} \text {s . t .} & \dot {h} (x, u) \geq - \alpha (h (x)). \end{array} \\ \end{array}
+$$
+
+Given the affine form of this constraint, this safety filter is a quadratic program (QP) that has a closed form solution (Ames et al., 2017) and can be solved fast enough for online safe controller synthesis (Gurrier et al., 2018).
+
+Since we are dealing with unknown systems (3) is not directly applicable, and we need to formulate a robust CBF constraint that accounts for potential modeling errors. To this end, we assume the CBF constraint is feasible up to a positive margin  $\epsilon > 0$ :
+
+Assumption 2.6 (Robust CBF Feasibility). There exists a radially unbounded, strictly monotonically increasing function  $\alpha$  with  $\alpha(0) = 0$  and a positive scalar  $\epsilon > 0$ , such that
+
+$$
+\sup  _ {u \in \mathcal {U}} \frac {\partial h}{\partial x} (x) (f (x) + g (x) u) \geq - \alpha (h (x)) + \epsilon \tag {5}
+$$
+
+holds for all  $x\in \mathcal{X}$
+
+Given the existence of a CBF, we note that this assumption is not very conservative. Intuitively, it states that we can violate the non-robust CBF by a margin of at most  $\epsilon$ .
+
+# 2.2. Gaussian Processes and Reproducing Kernel Hilbert Spaces
+
+We now introduce the tools used to model the unknown functions  $f$  and  $g$ .
+
+Consider a prior model of the system  $\hat{f}$  and  $\hat{g}$ , which can be set to zero if no prior knowledge is available. For the remainder of this paper, we assume  $\hat{f} = \hat{g} = 0$  for simplicity. We then model the residuals  $f - \hat{f}$  and  $g - \hat{g}$  jointly by using a GP model. To obtain theoretical guarantees on the growth of the model error as new data is added, we assume that the residual model belongs to a reproducing kernel Hilbert space (RKHS). This is codified later in this section. In the following, we review both GPs and RKHSs, and provide some preliminary theoretical results.
+
+A GP is a collection of random variables, of which any finite number is jointly Gaussian distributed. GPs are fully specified by a prior mean, which we set to zero without loss of generality, and kernel  $k: \mathcal{X} \times \mathcal{X} \to \mathbb{R}$ . We model each dimension of the unknown dynamics  $f(x) + g(x)u$  using a separate GP. This is achieved by employing composite kernels
+
+$$
+k _ {i} \left(\mathbf {z}, \mathbf {z} ^ {\prime}\right) \triangleq k _ {f _ {i}} \left(x, x ^ {\prime}\right) + \sum_ {j = 1} ^ {m} u _ {j} k _ {g _ {i, j}} \left(x, x ^ {\prime}\right) u _ {j} ^ {\prime}, \tag {6}
+$$
+
+where  $\mathbf{z} \triangleq (x^{\top}, u^{\top})^{\top}$ . The kernels  $k_{f_i}$  and  $k_{g_{i,j}}$  capture the behavior of the individual entries of the residuals  $f$  and  $g$ , respectively. In this paper, we do not require a specific form of kernel  $k_{f_i}$  and  $k_{g_{i,j}}$ , though the choice of kernel significantly affects performance. We employ a composite kernel (6) because this allows us to leverage measurements of  $\dot{x}$  to improve our learned model.
+
+Consider  $N$  noisy measurements  $\mathcal{D}_{i,N} \triangleq \{\mathbf{z}^{(q)}, y_i^{(q)}\}_{q=1,\dots,N}$  of the i-th entry of the time-derivative of the state, where
+
+$$
+y _ {i} ^ {(q)} = f _ {i} \left(x ^ {(q)}\right) + \sum_ {j = 1} ^ {m} g _ {i, j} \left(x ^ {(q)}\right) u _ {j} ^ {(q)} + \xi_ {i} ^ {(q)} \tag {7}
+$$
+
+and the measurement noise  $\xi^{(q)}$  satisfies the following assumption.
+
+Assumption 2.7. The measurement noise  $\xi^{(q)}$  is iid  $\sigma_{i,\mathrm{ns}}$ -sub Gaussian for every  $i\in \{1,\dots,n\}$ .
+
+The posterior of the model at an arbitrary state  $\mathbf{z}^*$  is then
+
+given by
+
+$$
+\begin{array}{l} \mu_ {i, N} (\mathbf {z} ^ {*}) \triangleq \mathbf {k} _ {i, *} ^ {\top} \left(\mathbf {K} _ {i, N} + \sigma_ {i, \mathrm {n s}} ^ {2} \mathbf {I} _ {N}\right) ^ {- 1} \mathbf {y} _ {i}, \\ \sigma_ {i, N} ^ {2} (\mathbf {z} ^ {*}) \triangleq k _ {i} (\mathbf {z} ^ {*}, \mathbf {z} ^ {*}) - \mathbf {k} _ {i, *} ^ {\top} \left(\mathbf {K} _ {i, N} + \sigma_ {i, \mathrm {n s}} ^ {2} \mathbf {I} _ {N}\right) ^ {- 1} \mathbf {k} _ {i, *}, \\ \end{array}
+$$
+
+where  $\mathbf{k}_{i,*}\triangleq (k_i(\mathbf{z}^*,\mathbf{z}^{(1)}),\dots,k_i(\mathbf{z}^*,\mathbf{z}^{(N)}))^\top$ , and the entries of the covariance matrix are given by  $[\mathbf{K}_{i,N}]_{pq} = k_i(\mathbf{z}^{(p)},\mathbf{z}^{(q)})$ . In our approach, we employ  $\mu_{i,N}$  as a model of our system, and  $\sigma_{i,N}$  as a measure of uncertainty, which we employ to inform the data-collection trigger.
+
+In the following, we refer to the mean and covariance matrix of the full multivariate Gaussian process model as
+
+$$
+\begin{array}{l} \begin{array}{l}\boldsymbol {\mu} _ {N} (\cdot) \triangleq \left(\mu_ {1, N} (\cdot), \dots , \mu_ {n, N} (\cdot)\right) ^ {\top},\\\overline {{\rightarrow 3}} (\cdot) \triangleq \left(3, \dots , 3, \dots , 3, \dots , 3\right)\end{array}\tag {8} \\ \boldsymbol {\Sigma} _ {N} ^ {2} (\cdot) \triangleq \operatorname {d i a g} \left(\sigma_ {1, N} ^ {2} (\cdot), \dots , \sigma_ {n, N} ^ {2} (\cdot)\right). \\ \end{array}
+$$
+
+Furthermore, we employ  $\mathcal{D}_N = \{\mathbf{z}^{(q)},\mathbf{y}^{(q)}\}_{q = 1,\dots ,N}$  where  $\mathbf{y}^{(q)}\triangleq (y_1^{(q)},\ldots ,y_n^{(q)})^\top$  to refer to the full data set corresponding to  $N$  system measurements.
+
+To be able to ensure safety, we need to estimate how data affects the worst-case model error. Though this is impossible in a general setting, an estimate is possible if we assume that the entries of  $f$  and  $g$  belong to an RKHS, a very rich function space specified by the composite kernel (6).
+
+Assumption 2.8. For every  $i = 1, \dots, n$ , the  $i$ -th entry of  $f(x) + g(x)u$  belongs to the RKHS with reproducing kernel  $k_{i}$ , and the corresponding RKHS norm is bounded by a known positive scalar  $B_{i} \in (0, \infty)$ .
+
+The satisfaction of Assumption 2.8 hinges on choosing the kernels correctly. In practice, if little is known about the system, then so-called "universal kernels" (e.g. Matérn or squared-exponential kernels) are often employed, which can approximate continuous functions arbitrarily accurately (Micchelli et al., 2006). Furthermore, there exist several techniques to mitigate kernel specification that can easily be applied to the methods described in this paper (Berkenkamp et al., 2019; Capone et al., 2022; Fiedler et al., 2021).
+
+# 3. Safe Control Through Bandit Exploration
+
+In this section, we present our main algorithm and theoretical guarantees.
+
+Our algorithms employs two key quantities: the lower confidence bound (LCB) and upper confidence bound (UCB) of the time derivative of  $h$  given the data. These quantities are
+
+computed as
+
+$$
+L C B _ {N} (x, u) = \frac {\partial h}{\partial x} (x) \boldsymbol {\mu} _ {N} (x, u) - L _ {h} \beta_ {N} \sqrt {\operatorname {t r} \left(\boldsymbol {\Sigma} _ {N} (x , u)\right)} \tag {9}
+$$
+
+$$
+U C B _ {N} (x, u) = \frac {\partial h}{\partial x} (x) \boldsymbol {\mu} _ {N} (x, u) + L _ {h} \beta_ {N} \sqrt {\operatorname {t r} \left(\boldsymbol {\Sigma} _ {N} (x , u)\right)}, \tag {10}
+$$
+
+where  $L_{h} > 0$  is the global Lipschitz constant of  $h$ , specified in Assumption 2.3 and
+
+$$
+\beta_ {N} \triangleq \max  _ {i \leq n} \left(B _ {i} + \sigma_ {i, \mathrm {n s}} \sqrt {2 \left(\gamma_ {i , N} + 1 + \log \left(n \delta^ {- 1}\right)\right)}\right). \tag {11}
+$$
+
+Here,  $\gamma_{i,N}$  denotes the maximal information gain after  $N$  rounds of data collection, and is computed as
+
+$$
+\gamma_ {i, N} = \max _ {\mathbf {z} ^ {(1)}, \dots \mathbf {z} ^ {(N)}} \frac {1}{2} \log \left| \mathbf {I} _ {N} + \sigma_ {i, \mathrm {n s}} ^ {- 1} \mathbf {K} _ {i, N} \right|.
+$$
+
+Intuitively, the LCB (9) and UCB (10) respectively correspond to the lowest and highest permissible value for  $\dot{h}(x,u)$  given the model error bound.
+
+Given the nominal controller  $\pi_{\mathrm{nom}}$ , we leverage the LCB (9) to compute robustly safe control inputs by solving the second-order cone program
+
+$$
+\begin{array}{l} \pi_ {N, \text {s a f e}} (x) = \underset {u \in \mathcal {U}} {\operatorname {a r g m i n}} \quad \| u - \pi_ {\text {n o m}} (x) \| _ {2} \\ \text {s . t .} \quad L C B _ {N} (x, u) \geq - \alpha (h (x)) + \frac {\epsilon}{2}. \tag {12} \\ \end{array}
+$$
+
+Note that, due to the nature of the composite kernels (6),  $\pmb{\mu}_N(x,u)$  is a linear function of  $u$  and  $\Sigma_N(x,u)$  is a diagonal matrix whose entries are positive definite quadratic functions of  $u$ . Hence, (12) can be solved straightforwardly with conventional second-order cone program optimizers.
+
+We include the term  $\frac{\epsilon}{2}$  in the constraint in (12) because this keeps the system from getting too close to the boundary. This ensures that when (12) becomes infeasible, there is still some distance to go until it reaches the boundary of the safe set, leaving us with enough time to explore the system.
+
+As long as the robust safety filter (12) is strictly feasible, the resulting control policies are locally Lipschitz continuous (Castañeda et al., 2022), yielding unique and well-behaved closed-loop system trajectories.
+
+If (12) is feasible for all  $x \in \mathcal{C}$ , we can leverage standard results from CBF theory (Ames et al., 2017) to show that the resulting trajectory is safe. However, since (12) includes model uncertainty, it becomes potentially infeasible in regions of the state space where uncertainty is high. We address this by employing a bandit-type exploration algo
+
+rithm to acquire new data whenever infeasibility occurs. Our algorithm is designed to collect sufficiently informative data to render (12) feasible before exiting the safe set. We now describe the corresponding sampling scheme in detail.
+
+Let  $N$  denote the number of data collected up to the current time. If (12) is strictly feasible, it is used to compute control inputs. As soon as strict feasibility is impossible, we switch to our bandit exploration strategy. This corresponds to the first instance in time when
+
+$$
+\max  _ {u \in \mathcal {U}} L C B _ {N} (x, u) = - \alpha (h (x)) + \frac {\epsilon}{2} \tag {13}
+$$
+
+holds and can be easily verified using a quadratic program. We denote this point in time by  $t_{N + 1}$ . At this point, we switch to an exploration-driven controller  $\bar{\pi}_{N + 1}$ , which can be chosen freely, provided that it is locally Lipschitz continuous and satisfies  $\bar{\pi}(x(t_{N + 1})) = u^{(N + 1)}$ , where  $u^{(N + 1)}$  maximizes the upper confidence bound
+
+$$
+u ^ {(N + 1)} = \arg \max  _ {u \in \mathcal {U}} U C B _ {N} (x (t _ {N + 1}), u). \tag {14}
+$$
+
+In our experiments in Section 5, we choose  $\bar{\pi}(x) \equiv u^{(N+1)}$ . Choosing a control input this way is very similar to bandit exploration algorithms, which aim to find the maximum of an unknown function. In our case, we wish to maximize the unknown time derivative of the CBF  $\dot{h}(x,u)$  given the state  $x$ , which we know to satisfy Assumption 2.6. We apply  $\bar{\pi}_{N+1}$  for a time length of  $\Delta t$ , where  $\Delta t$  corresponds to the data sampling frequency and is specified a priori. We then collect a noisy measurement of the time-derivative of the system
+
+$$
+\mathbf {y} ^ {(N + 1)} = f (x (t _ {N + 1})) + g (x (t _ {N + 1})) u ^ {(N + 1)} + \xi^ {(N + 1)}
+$$
+
+and update the GP model with the measurement data pair  $(\mathbf{z}^{(N + 1)},\mathbf{y}^{(N + 1)})$ , where
+
+$$
+\mathbf {z} ^ {(N + 1)} = \left(\left(x (t _ {N + 1})\right) ^ {\top}, \left(u ^ {(N + 1)}\right) ^ {\top}\right) ^ {\top}.
+$$
+
+After a time of  $\Delta t$  has passed, we check if the optimization problem (12) is strictly feasible. To this end, we check if
+
+$$
+\max  _ {u \in \mathcal {U}} L C B _ {N} (x, u) > - \alpha (h (x)) + \frac {\epsilon}{2} \tag {15}
+$$
+
+holds. If so, we apply  $\pi_{N,\text{safe}}$ . Otherwise, we repeat the sampling procedure. These steps are summarized in Algorithm 1.
+
+Remark 3.1. The condition (13) specifies that exploration starts when strict infeasibility is not given, as opposed to waiting until infeasibility occurs. In general, it is not advisable to wait for infeasibility to start exploring, as this means that the control law is not well-defined, which can be
+
+Algorithm 1 Safe Control via On-the-Fly Bandit Exploration
+
+Input: Sampling time  $\Delta t$ , GP prior, CBF  $h$ , class- $\mathcal{KL}$  function  $\alpha$   
+1: Set EXPLORE = FALSE  
+2: for  $t \in [0, \infty)$  do  
+3: if EXPLORE == FALSE then  
+4: if  $\max_{u \in \mathcal{U}} LCB_N(x, u) > -\alpha(h(x)) + \frac{\epsilon}{2}$  then  
+5: Solve (12) and apply  $\pi_{N,\text{safe}}(x)$   
+6: else  
+7: Set EXPLORE = TRUE  
+8: Set  $N = N + 1$ .  
+9: Set  $t_N = t$ .  
+10: Set  $x^{(N)} = x$ .  
+11: Compute  $u^{(N)}$  by solving (14).  
+12: end if  
+13: end if  
+14: if EXPLORE == TRUE then  
+15: if  $t < t_N + \Delta t$  then  
+16: Apply a locally Lipschitz controller  $\bar{\pi}$  with  $\bar{\pi}(x^{(N)}) = u^{(N)}$ .  
+17: Collect noisy measurement  $\mathbf{y}^{(N)} = \dot{x}(t_N) - \hat{f}(x(t_N)) - \hat{g}(x(t_N))u^{(N)} + \xi^{(N)}$   
+18: else if  $t = t_N + \Delta t$  then  
+19: Set EXPLORE = FALSE  
+20: Set  $\mathcal{D}_N = \mathcal{D}_{N-1} \cap \{\mathbf{z}^{(N)}, \mathbf{y}^{(N)}\}$  and update GP.  
+21: end if  
+22: end if  
+23: end for
+
+detrimental to control.
+
+We then show that, if we choose  $\Delta t$  high enough, the closed-loop system under Algorithm 1 is safe with high probability.
+
+Theorem 3.2. Let Assumptions 2.1, 2.3, 2.4 and 2.6 to 2.8 hold. Choose  $\delta \in [0,1]$  and  $\beta_{N}$  as in (11). Moreover, choose the sampling time as
+
+$$
+\Delta t > \frac {\epsilon}{L _ {\alpha} L _ {h} L _ {\dot {x}} \Delta N _ {\max }}, \tag {16}
+$$
+
+where  $\Delta N_{\max}$  satisfies
+
+$$
+\Delta N _ {\max } > \frac {3 2 \beta_ {\Delta N _ {\max }} ^ {2} L _ {h} ^ {2}}{\epsilon^ {2} \log \left(1 + \sigma_ {i , n s} ^ {- 2}\right)} \sum_ {i = 1} ^ {n} \gamma_ {i, \Delta N _ {\max }} \tag {17}
+$$
+
+Let the initial state  $x(0)$  be strictly within the safe set, such that  $\alpha(h(x(0))) \geq \epsilon$ . Then, with probability at least  $1 - \delta$ , the closed-loop system under the policy specified by Algorithm 1 is safe. Furthermore, the closed-loop stops collecting data after at most  $\Delta N_{\text{max}}$  collected observations.
+
+The proof of Theorem 3.2 can be found in Appendix B.
+
+![](images/5d71d337d5ff2f50a6bda7c5efb7aae1038ceb94ad977ac91959d8de6df6c12b.jpg)  
+(a) Value of control barrier function  $h(x)$  for cruise control example. Since no prior model is available for control, safety is obtained by efficiently learning a model online.
+
+![](images/a6b7300102290987356a56404d2fb0a1e9a740bc922e5b1b0af6c72811d7b9a4.jpg)  
+(b) Minimum of LCB of CBF time derivative for cruise control example. Spikes are due to training data set updates, leading to decreased model uncertainty. Positive values indicate infeasibility of the robust safety filter (12).  
+Figure 2. CBF and LCB for a single cruise control simulation using a sampling time of  $\Delta t = 10^{-5}\mathrm{s}$ .
+
+Intuitively, Theorem 3.2 states that if we choose the data collection frequency high enough, we guarantee safety after collecting a finite number of data. The quantity  $\beta_{\Delta N_{\mathrm{max}}}^2 \sum_{i=1}^{n} \gamma_i, \Delta N_{\mathrm{max}}$  is the only part in the right-hand side of (17) that depends on  $\Delta N_{\mathrm{max}}$ . It is closely related to the regret metric frequently found in bandit literature, which measures the deviation from optimal decisions. This quantity grows sublinearly for most commonly employed kernels, meaning there exists a  $\Delta N_{\mathrm{max}}$  that satisfies (17). In practice, we can approximate by sampling the state and input space. Alternatively, we can obtain a (potentially crude) bound for this term based on the kernel hyperparameters. This is discussed in Appendix C.
+
+# 4. Discussion and Limitations
+
+Although we assume that the system structure (1) is affine in the control input, this is not strictly a requirement for theoretical guarantees and can be relaxed, provided that the optimization problems (12), (13) and (14) can be solved adequately.
+
+Theorem 3.2 points at an interaction between various quantities, including the sampling frequency, model uncertainty, measurement noise, CBF violation tolerance, and Lipschitz constants of the system and CBF. In simple terms, Theorem 3.2 indicates that a higher sampling frequency is likely more beneficial for safety. However, other practical considerations can also improve safety. For instance, we can reduce the Lipschitz constant  $L_{\hat{x}}$  of the system by cleverly restricting the permissible control input space  $\mathcal{U}$ , thereby reducing the bound (17).
+
+A practical concern that may arise with control inputs  $u$  geared towards exploration is that the corresponding inputs may strain the system and lead to undesirable behavior, e.g., if the input exhibits a high frequency and amplitude. However, computing the input by maximizing the UCB
+
+(14) corresponds to choosing the optimistically safest input under uncertainty. Hence, it is reasonable to expect the corresponding input to be acceptable to the system, i.e., not damaging.
+
+The primary focus of this work is the safety of the closed-loop function, and we do not address closed-loop control performance. However, the tools presented here can be straightforwardly extended to address performance, e.g., by designing a data collection scheme that aims to reduce model uncertainty whenever control performance is not satisfactory. Furthermore, our method can be potentially improved by preemptively exploring before infeasibility occurs, increasing the space of permissible control actions.
+
+Although Assumption 2.6 can seem strict, it still offers flexibility, and there are ways to relax it in practice. Typically, only a portion of the safe set is visited during control. Hence, it is enough if the CBF is valid only for the corresponding subset of the state space. Furthermore, we can achieve conservatism by initially restricting the safe set to a region that is easy to control and then iteratively expanding it as more data is collected. This potentially allows us to gradually improve the CBF as more system knowledge allows us to expand the safe set. Alternatively, we can include conservatism in the safety requirement by computing the CBF condition from a collection of (potentially valid) CBFs and taking the worst case.
+
+# 5. Numerical Experiments
+
+We now showcase how our approach performs using two numerical simulations: a cruise control system and a quadrotor with ground dynamics. We assume to have either no prior model (cruise control example) or to know only the model component corresponding to the time derivatives (quadrotor). Note that this is insufficient to implement any state-of-the-art approach, as the prior model does not include
+
+![](images/27382bb69d3ef008956b152537c172c9234b2fb9027686d7e4cd3729bb68de76.jpg)  
+(a) The quadrotor starts at a state within the safe region where (12) is infeasible.
+
+![](images/18bd36d1041a30c996ebdd5d45841dc0e0b27c693c2e0e204391c6b7b67e4b39.jpg)  
+(b) Our algorithm computes exploratory control inputs without leaving the safe set. The observations are used to update the model. This phase corresponds roughly to the first two seconds of the simulation.
+
+![](images/2695a793d11fbd8a43b03032bd4c181dd4d0082067bca600d0e350140b50da4d.jpg)  
+(c) After a finite amount of data is collected without exiting the safe set, the uncertainty is small enough to achieve the feasibility of the safety filter (12). This is expected from Theorem 3.2.
+
+![](images/b1f21c9baf9ad4a81e57583185f47722d582c01bfd5021e9aa0f5032b7b38e1a.jpg)  
+Figure 3. Visualization of a single trajectory of the quadrotor experiment using a sampling time of  $\Delta t = 10^{-5}\mathrm{s}$ .  
+(a) Value of control barrier functions for quadrotor example.  
+Figure 4. CBF and LCB for a single quadrotor run using a sampling time of  $\Delta t = 10^{-5}\mathrm{s}$ .
+
+![](images/412be21839878c3ab85ef82692a18140d9dfb3c26b7d879e8232725e144feac6.jpg)  
+(b) Minimum of the LCB of the CBF time derivative for quadrotor example.
+
+the influence of the control input. We additionally perform experiments that aim to answer the following questions:
+
+- How does our method compare with random exploration during infeasibility?  
+- How does the choice of sampling time  $\Delta t$  affect safety?
+
+# 5.1. Cruise Control
+
+We employ our approach to learn the road vehicle model presented in Castañeda et al. (2021) while applying an adaptive cruise control system. The states are given by  $x = [v z]^\top$ , where  $z$  is the distance between the ego vehicle and the target vehicle in front, and  $v$  denotes the ego vehicle speed. Our setup is fully described in Appendix A.1.
+
+As a control barrier function, we employ  $h(x) = z - T_{h}v$ , where  $T_{h} = 1.8$ , which aims to maintain a safe distance between the ego vehicle and the vehicle in front. The nominal controller  $\pi_{\mathrm{nom}}(x)$  used for the robust CBF-SOCP (12) is a P-controller  $\pi_{\mathrm{nom}} = -10(v - v_d)$ , where  $v_{d} = 24$
+
+corresponds to the desired velocity. We employ squared-exponential kernels to model  $f$  and  $g$  and assume to have  $N = 10$  data points at the start of the simulation, which we employ exclusively to learn the kernel hyperparameters. We do so by minimizing the posterior likelihood (Rasmussen & Williams, 2006).
+
+We simulate the system for 100 seconds. The CBF value for a single simulation run using a sampling time of  $\Delta t = 10^{-5}\mathrm{s}$  is shown in Figure 2a. The minimum of the LCB is depicted in Figure 2b. The CBF value  $h(x)$  is always above zero, meaning the system is safe. This is to be expected from Theorem 3.2. As can be seen in Figure 2b, the minimum of the LCB is frequently positive in the beginning of the simulation. This means that the safety filter (12) is infeasible during these time instances, particularly in the beginning. This leads to a high rate of exploratory inputs, which suddenly decreases model uncertainty, causing the LCB to spike. After approximately 6 seconds, enough information was collected to recover feasibility, after which a safe input can be obtained without further exploration.
+
+![](images/a4f439b903ef8e286e3bbff7f2850883c7b9aa567d712387b62455c2e1fedaf4.jpg)  
+Figure 5. Rate of failure using our approach and random inputs for the cruise control (left) and quadrotor (right) settings at varying sampling frequencies  $(\Delta t)^{-1}$ .
+
+![](images/0514a68f0315b0b145e60b2d81b93ea6e0bb715ea08c86b660c7823bfcfc722f.jpg)
+
+# 5.2. Quadrotor
+
+We additionally showcase our approach using a numerical simulation of a quadrotor with ground dynamics. The quadrotor dynamics are described in detail in Appendix A.2.
+
+The quadrotor states are given by  $x = [p, v, q]$ , where where  $p \in \mathbb{R}^3$  is the global position,  $v \in \mathbb{R}^3$  the global velocity, and  $q \in \mathbb{R}^4$  is the system orientation in quaternion format. The control inputs  $u = [T\omega]^{\top}$  consist of the thrust acceleration  $T$  and the body-frame angle rates  $\omega \in SO(3)$ .
+
+In this example, we employ two CBFs. The first is  $h(x) = 10(p_z - T_zv_z)$ , where  $T_{z} = 0.1$ , and is geared toward keeping altitude higher than zero. The second CBF utilizes the differential flatness of the quadrotor to restrict velocities, orientation we add a rotation, and the orientation of the thrust vector. It is fully described in Appendix A.2. When computing the exploratory control inputs (14), we alternate between CBFs. The nominal controller  $\pi_{\mathrm{nom}}(x)$  used for the robust safety filter (12) corresponds to a differentially flat controller, computed as in Faessler et al. (2018), and we consider bounded thrust, with  $|T| \leq 15000$ . Similarly to the cruise control setting, we use squared-exponential kernels and assume to have  $N = 10$  data points at the start of the simulation to learn the kernel hyperparameters.
+
+We simulate the system for 50 seconds. The CBF value for a single run using a sampling time of  $\Delta t = 10^{-5}\mathrm{s}$  can be seen in Figure 4a, the minimum of the LCB is depicted in Figure 4b. Figure 4 depicts the corresponding quadrotor trajectory. Similarly to the cruise control setting, infeasibility is frequently encountered initially, causing our algorithm to explore. After approximately 2 seconds, feasibility is recovered, causing the quadrotor to hover in place for the remainder of the simulation.
+
+# 5.3. Comparison with Random Exploratory Control
+
+Theorem 3.2 states that it is sufficient to collect data by applying (14) to the system in order to guarantee safety. However, other types of control inputs may also satisfy this requirement. In the following, we investigate how our approach performs compared to random exploratory control inputs. More specifically, we apply our method to the system with the following difference: instead of computing exploratory inputs by solving (14), we sample the control inputs from a uniform distribution on  $\mathcal{U}$ . We again consider the adaptive cruise control and quadrotor settings and investigate how our approach and the random input-based one perform using various sampling frequencies  $\Delta t$ . This is especially relevant, as many practical settings do not allow for arbitrarily high sampling frequencies.
+
+We perform 100 simulations with different initial conditions, uniformly sampled from a region within the safe set. We report how often each method fails, i.e., leads to a positive value for the CBF during the simulation. The average number of failures for both settings is shown in Figure 5. A non-zero failure rate is expected at low sampling frequencies since too little data is collected to learn a model quickly. However, our approach nonetheless performs better than the random control input-based approach. This is because the inputs applied to the system are geared towards recovering the feasibility of the robust safety filter (12), whereas random inputs are not. We also observe that data efficiency is higher at higher sampling, leading to less collected data. This is because a higher sampling rate means that the elapsed time between data collection and model update is smaller, i.e., the model captures the true system more faithfully immediately after an update. This suggests that, if the maximal permissible sampling frequency is low, then particular effort should be put into updating the model as fast as possible.
+
+# 6. Related Work
+
+# 6.1. Control Barrier Functions
+
+CBFs have been studied extensively in settings with perfectly known dynamics (Agrawal & Sreenath, 2017; Ames et al., 2017; Bansal et al., 2017; Wabersich & Zeilinger, 2018; Zeng et al., 2021). However, the setting where a known CBF is used to guide data collection when dynamics are unknown is an ongoing research question that has recently garnered increased attention. In (Choi et al., 2020; Taylor et al., 2020), the authors employ a neural network to learn the residual dynamics in a reinforcement learning setting. The safety of the learned policy is enforced by including a CBF constraint during training. An event-triggered approach for improving the learned dynamics model is proposed in (Lederer et al., 2024), where a CBF and the system structure are leveraged to guarantee safety.
+
+# 6.2. Safe Exploration with Gaussian Processes
+
+Gaussian processes have been used extensively in the context of safe control and exploration. The work of (Sukhija et al., 2023) leverages contextual bandits to safely tune a parametric controller. By formulating an upper confidence bound for the safety function, the method switches to safe parameters whenever the boundary of the safe set is reached. The works of Prajapat et al. (2022); Sui et al. (2015); Turchetta et al. (2016); Wachi & Sui (2020) consider exploration with finite state spaces, known dynamics, and unknown functions specifying safe state-action pairs. A lower confidence bound of the safety constraints ensures safe actions, whereas an upper confidence bound for the reward drives exploration. A significant difference between our method and those described above is that they explore safe states to expand the safe set and maximize return, whereas exploration in our approach is driven by the need to ensure safety. The methods of (Hewing et al., 2019; Koller et al., 2018) use a GP to formulate safety constraints over a finite horizon for a model predictive control algorithm.
+
+# 7. Conclusion
+
+We have presented a learning-based control approach for exploring on the fly to guarantee safety. Using a Gaussian process model, we have formulated a robust safety filter that guarantees safe inputs whenever feasible. Whenever infeasibility occurs we revert to a bandit exploration algorithm that is geared toward ensuring that feasibility is recovered before exiting the safe set. We show that if the sampling rate is chosen high enough, we can provably ensure safety with high probability. In future, we aim to apply the proposed approach to real-life and other complex systems.
+
+# Acknowledgements
+
+This work is supported by the DAAD programme Kon- rad Zuse Schools of Excellence in Artificial Intelligence (relAI), sponsored by the German Federal Ministry of Education and Research and by the European Research Council (ERC) Consolidator Grant "Safe data-driven control for humancentric systems (CO-MAN)" under grant agreement number 864686.
+
+# Impact Statement
+
+This paper presents work toward advancing the safety of Machine Learning-based methods. Safety plays a crucial role in developing Machine Learning techniques, as guaranteeing safety greatly enhances the applicability of existing methods. We feel that developing and further improving safe learning-based techniques such as the one presented in this paper will have a significant societal impact, as the spectrum of applications is substantial, including healthcare, autonomous driving, human-machine interaction, and various other robotics fields.
+
+# References
+
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+
+# A. Full Description of Numerical Experiments
+
+# A.1. Cruise Control
+
+In the following, we omit physical dimensions when describing the system model. The state space model of the cruise control is given by  $\dot{x} = f(x) + g(x)u$ , with state-dependent functions
+
+$$
+f (x) = \left[ \begin{array}{c} - \frac {1}{m} \left(\zeta_ {0} + \zeta_ {1} v + \zeta_ {2} v ^ {2}\right) \\ v _ {0} - v \end{array} \right], \quad g (x) = \left[ \begin{array}{l} 0 \\ \frac {1}{m} \end{array} \right] \tag {18}
+$$
+
+and state  $x = [v z]^\top$ , where  $z$  denotes the distance between the ego vehicle and the target vehicle in front,  $v$  denotes the ego vehicle speed,  $m = 1650$  its mass, and  $\zeta_0 = 0.2$ ,  $\zeta_1 = 10$ ,  $\zeta_2 = 0.5$  are parameters that specify the rolling resistance.
+
+# A.2. Quadrotor
+
+The quadrotor dynamics are specified by the functions
+
+$$
+\dot {p} = v, \quad \dot {v} = g _ {\mathrm {g r}} e _ {z} + \zeta (p _ {z}) R e _ {z} T, \tag {19a}
+$$
+
+$$
+\dot {R} = R [ \omega ] _ {\times} \tag {19b}
+$$
+
+where  $p \in \mathbb{R}^3$  is the global position,  $v \in \mathbb{R}^3$  the global velocity, and  $R \in \mathrm{SO}(3)$  the system orientation. The parameter  $g_{\mathrm{gr}} = 9.81$  denotes gravity,  $[\cdot]_{\times} : \mathbb{R}^3 \to \mathbb{R}^{3 \times 3}$  is the skew-symmetric mapping, and  $e_z = [001]^\top$  is the unit- $z$  vector. The function  $\zeta : \mathbb{R}_+ \to [0,1]$  models ground effects and takes the quadrotor height  $p_z$  as an input variable. It is computed as Danjun et al. (2015)
+
+$$
+\zeta \left(p _ {z}\right) = 1 - \rho \left(\frac {r _ {\mathrm {r o t}}}{4 p _ {z}}\right) ^ {2}, \tag {20}
+$$
+
+where  $\rho = 5$  and  $r_{\mathrm{rot}} = 0.09$  is the rotor radius.
+
+The first CBF is  $h(x) = 10(p_z - T_zv_z)$ , where  $T_{z} = 0.1$ . It is geared toward keeping altitude higher than zero. The second CBF utilizes the differential flatness of the quadrotor to synthesize a viable safe set. In particular, we are concerned with restricting the system to safe positions defined as the 0-superlevel set of  $h_p(p) = r^2 -\| p\| ^2$ . We extend  $h_p$  to include velocities as  $h_e(p,v) = \dot{h}_p(p,v) + \alpha h_p(p)$  for some  $\alpha >0$  to produce a relative degree-1 CBF for a double integrator system as in Nguyen & Sreenath (2016). To include orientation we add a rotation term to produce a CBF for the drone,  $h(p,v,R) = h_e(p,v) - \lambda (1 - \frac{1}{2r}\frac{\partial h}{\partial p} Re_z)$ , with some  $\lambda \in (0,r^2 /2)$  which shrinks the safe set to ensure that the thrust vector is pointing inwards whenever  $h_e(p,v) = 0$ .
+
+# B. Proof of Theorem 3.2
+
+We prove Theorem 3.2 in two steps. First, we show that the control inputs are well-behaved and the closed-loop system is safe if the robust CBF-SOCP (12) is strictly feasible. Afterward, we show that strict feasibility is always recovered before exiting the safe set when using our exploration approach.
+
+We begin by leveraging Assumptions 2.7 and 2.8 to obtain a bound on the model error, which is crucial to establish safety.
+
+Lemma B.1. Let Assumptions 2.7 and 2.8 hold and let
+
+$$
+\beta_ {N} \triangleq \max  _ {i \leq n} \left(B _ {i} + \sigma_ {i, n s} \sqrt {2 \left(\gamma_ {i , N} + 1 + \log \left(n \delta^ {- 1}\right)\right)}\right).
+$$
+
+Then, with probability at least  $1 - \delta$
+
+$$
+\left\| f \left(x ^ {*}\right) + g \left(x ^ {*}\right) u ^ {*} - \boldsymbol {\mu} \left(x ^ {*}, u ^ {*}\right) \right\| _ {2} \leq \beta_ {N} \sqrt {\operatorname {t r} \left(\boldsymbol {\Sigma} _ {N} \left(x ^ {*} , u ^ {*}\right)\right)}
+$$
+
+holds for all  $x\in \mathcal{X}$ $u\in \mathcal{U}$  and all  $N\in \mathbb{N}$
+
+Proof. By Capone & Hirche (2019, Lemma 1), with probability at least  $1 - \delta$ ,
+
+$$
+\left| f _ {i} \left(x ^ {*}\right) + \sum_ {j = 1} ^ {m} g _ {i, j} \left(x ^ {*}\right) u _ {j} ^ {*} - \mu_ {i, N} \left(x ^ {*}, u ^ {*}\right) \right| \leq \left(B _ {i} + \sigma_ {i, \mathrm {n s}} \sqrt {2 \left(\gamma_ {i , N} + 1 + \log \left(n \delta^ {- 1}\right)\right)}\right) \sigma_ {i, N} \left(x ^ {*}, u ^ {*}\right)
+$$
+
+holds for all  $i = 1,\dots,n$ , all  $x\in \mathcal{X}$ ,  $u\in \mathcal{U}$ , and all  $N\in \mathbb{N}$ . We then have
+
+$$
+\begin{array}{l} \left\| f \left(x ^ {*}\right) + g \left(x ^ {*}\right) u ^ {*} - \boldsymbol {\mu} \left(x ^ {*}, u ^ {*}\right) \right\| _ {2} ^ {2} \leq \sum_ {i = 1} ^ {n} \left| f _ {i} \left(x ^ {*}\right) + \sum_ {j = 1} ^ {m} g _ {i, j} \left(x ^ {*}\right) u _ {j} ^ {*} - \mu_ {i, N} \left(\mathbf {z} ^ {*}\right) \right| ^ {2} \\ \leq \sum_ {i = 1} ^ {n} \beta_ {i, N} ^ {2} \sigma_ {i, N} ^ {2} (\mathbf {z} ^ {*}) \\ \leq \sum_ {i = 1} ^ {n} \beta_ {N} ^ {2} \sigma_ {i, N} ^ {2} (\mathbf {z} ^ {*}) \\ = \beta_ {N} ^ {2} \operatorname {t r} \left(\boldsymbol {\Sigma} _ {N} \left(x ^ {*}, u ^ {*}\right)\right). \\ \end{array}
+$$
+
+![](images/cd51f73ded551ebb9a7f0f6f36dff69116dc2ef81dab56eb9b09e5a35cd4d25d.jpg)
+
+Lemma B.1 allows us to bound the GP model error at any given point  $\mathbf{z}^*$  given any data with high probability. This plays a crucial role in establishing robust system safety in the face of model uncertainty.
+
+# Safety under Strict Feasibility
+
+We first demonstrate that when the robust safety filter (12) is strictly feasible, the control law  $\pi_{N,\mathrm{safe}}$  ensures the system remains strictly within the safe set with high probability. Specifically, the control barrier function value of the closed-loop system increases over time whenever  $\alpha(h(x)) < \frac{\epsilon}{2}$ . To prove this statement formally, we first present some preliminary results.
+
+Lemma B.2. Let the robust safety filter (12) be strictly feasible for some  $x \in \mathcal{X}$ . Then the policy  $\pi_{N, safe}$  specified by the safety filter (12) is Lipschitz continuous in a neighborhood of  $x$ .
+
+Proof. The proof is identical to that of Castaneda et al. (2022, Lemma 6).
+
+Lemma B.2 is important to guarantee that the trajectory resulting from applying  $\pi_{N,\mathrm{safe}}$  is unique and well-behaved, allowing us to establish safety.
+
+Lemma B.3. Let Assumptions 2.1, 2.3 and 2.6 to 2.8 hold. Then, with probability at least  $1 - \delta$ , the following holds for all  $x \in \mathcal{X}$ , all  $u \in \mathcal{U}$ , and all  $N \in \mathbb{N}$ :
+
+$$
+L C B _ {N} (x, u) \leq \dot {h} (x, u) \leq U C B _ {N} (x, u). \tag {21}
+$$
+
+Proof. We show the result only for the lower bound  $LCB_{N}$  since the argument is identical for the upper bound  $UCB_{N}$ . By
+
+Lemma B.1, with probability at least  $1 - \delta$
+
+$$
+\begin{array}{l} \dot {h} (x, u) = \frac {\partial h}{\partial x} (x) (f (x) + g (x) u) \\ = \frac {\partial h}{\partial x} (x) (\boldsymbol {\mu} _ {N} (x, u)) + \frac {\partial h}{\partial x} (x) (f (x) + g (x) u - \boldsymbol {\mu} _ {N} (x, u)) \\ \geq \frac {\partial h}{\partial x} (x) (\boldsymbol {\mu} _ {N} (x, u)) - \left\| \frac {\partial h}{\partial x} (x) \right\| _ {2} \| (f (x) + g (x) u - \boldsymbol {\mu} _ {N} (x, u)) \| _ {2} \\ \geq \frac {\partial h}{\partial x} (x) (\boldsymbol {\mu} _ {N} (x, u)) - \left\| \frac {\partial h}{\partial x} (x) \right\| _ {2} \beta_ {N} \sqrt {\operatorname {t r} \left(\boldsymbol {\Sigma} _ {N} (x , u)\right)} + \frac {\epsilon}{2} \\ \geq \frac {\partial h}{\partial x} (x) (\boldsymbol {\mu} _ {N} (x, u)) - L _ {h} \beta_ {N} \sqrt {\operatorname {t r} \left(\boldsymbol {\Sigma} _ {N} (x , u)\right)} \\ = L C B _ {N} (x, u). \\ \end{array}
+$$
+
+holds for all  $x\in \mathcal{X}$  ,all  $u\in \mathcal{U}$  ,and all  $N\in \mathbb{N}$
+
+![](images/56b6f39ba61290dcb6196a6448a931cf422959d613239d908247fc16a2373a56.jpg)
+
+We additionally require the following result, which is a direct consequence of a classical result on CBFs (Ames et al., 2017, Theorem 1).
+
+Lemma B.4. For some  $\tau_{max} \in \mathbb{R} \cup \infty$ , let  $\tilde{x}(t)$ ,  $t \in [0, \tau_{max})$ , be the unique and well-defined solution to the differential equation
+
+$$
+\dot {\tilde {x}} = \tilde {f} (\tilde {x}), \quad \tilde {x} \in \mathcal {X}, \quad \tilde {x} (0) = \tilde {x} _ {0} \tag {22}
+$$
+
+where  $\tilde{f}$  is Lipschitz continuous around a neighborhood of  $\tilde{x}(t)$  for all  $t \in [0, \tau_{\max})$ . If  $h(\tilde{x}(0)) > 0$  and
+
+$$
+\frac {\partial h}{\partial \tilde {x}} (\tilde {x} (t)) \tilde {f} (\tilde {x} (t)) \geq - \alpha (h (\tilde {x} (t))) + \frac {\epsilon}{2} \tag {23}
+$$
+
+holds for all  $t\in [0,\tau_{\mathrm{max}})$  , then  $\alpha (h(\tilde{x} (t))) > 0$  for all  $t\in [0,\tau_{\mathrm{max}})$
+
+Proof. The proof is identical to that of Ames et al. (2017, Theorem 1). Note that, although Ames et al. (2017, Theorem 1) requires  $\tilde{f}$  to be locally Lipschitz continuous in  $\mathcal{X}$  and (23) to hold for any state in  $\mathcal{X}$ , these properties are only used to generalize the result to all solutions of (22) with arbitrary  $\tilde{x}(0) \in \mathcal{X}$ , which we do not require in this case.
+
+We now show that finite-time trajectories obtained with the controller  $\pi_{N,\text{safe}}$  are always strictly within the safe set, provided that the safety filter (12) is strictly feasible.
+
+Lemma B.5. Let Assumptions 2.1, 2.3 and 2.6 to 2.8 hold. For all  $N \in \mathbb{N}$ , define
+
+$$
+\mathcal {X} _ {N} = \left\{x \in \mathcal {X} \left| \max _ {u \in \mathcal {U}} L C B _ {N} (x, u) > - \alpha (h (x)) + \frac {\epsilon}{2} \right. \right\}
+$$
+
+and the dynamical system
+
+$$
+\dot {x} _ {N} = f \left(x _ {N}\right) + g \left(x _ {N}\right) \pi_ {\text {s a f e}, N} \left(x _ {N}\right), \quad t \in \left[ t _ {N} + \Delta t, t _ {N + 1}\right), \tag {24}
+$$
+
+specified within the time interval  $[t_N + \Delta t, t_{N+1})$ , where the initial condition satisfies  $x_N(t_N + \Delta t) \in \mathcal{X}_N$ ,  $t_{N+1}$  is given by
+
+$$
+t _ {N + 1} \triangleq \inf  \left\{t \geq t _ {N} + \Delta t \left| \max  _ {u \in \mathcal {U}} L C B _ {N} (x, u) = - \alpha (h (x)) + \frac {\epsilon}{2} \right. \right\}, \tag {25}
+$$
+
+and  $\alpha(h(x_{N}(t_{N} + \Delta t))) > 0$ , and  $\pi_{safe,N}$  is given by (12). Then, if  $\mathcal{X}_{N}$  is non-empty, (24) has a unique solution  $x_{N}(t)$ . Furthermore, with probability at least  $1 - \delta$ , the trajectories  $x_{N}(t)$  satisfy the following conditions for all  $N \in \mathbb{N}$ :
+
+(i)  $x_{N}(t)\in \mathcal{C}$  for  $t\in [t_N + \Delta t,t_{N + 1})$  
+(ii) If  $\alpha(h(x_N(\hat{t}))) < \frac{\epsilon}{2}$  for any  $\hat{t} \in [t_N + \Delta t, t_N)$ , then  $\alpha(h(x_N(t))) > \alpha(h(x_N(\hat{t})))$  for all  $t \in (\hat{t}, t_N)$ .  
+(iii) If  $\alpha\left(h(x_{N}(\hat{t}))\right) \leq \frac{\epsilon}{2}$  for any  $\hat{t} \in [t_{N} + \Delta t, t_{N}]$ , then  $\alpha\left(h(x_{N}(t))\right) \geq \alpha\left(h(x_{N}(\hat{t}))\right)$  for all  $t \in [\hat{t}, t_{N}]$ .
+
+Proof. We begin by showing condition (i), then show (ii) and (iii).
+
+(i) Due to the definition of  $\mathcal{X}_N$ , the safety filter (12) is strictly feasible for all  $x \in \mathcal{X}_N$ . Furthermore, the definition of  $t_{N+1}$  implies that a candidate solution  $x_N(t)$  satisfies  $x_N(t) \in \mathcal{X}_N$  for all  $t \in [t_N + \Delta t, t_{N+1})$ . Moreover,  $\pi_{N,\text{safe}}$  is well-defined and strictly feasible in  $\mathcal{X}_N$ . By Lemma B.2,  $\pi_{N,\text{safe}}$  is Lipschitz continuous around a neighborhood of any  $x \in \mathcal{X}_N$ . Since  $f$  and  $g$  are locally Lipschitz continuous, this implies that  $f(x) + g(x)\pi_{N,\text{safe}}(x)$  is locally Lipschitz continuous in  $\mathcal{X}_N$ . Hence, by the Picard-Lindelöf Theorem, (24) has a unique solution  $x_N(t)$ . Since  $\mathcal{X}_N$  is bounded and  $x_N(t) \in \mathcal{X}_N$  for all  $t \in [t_N + \Delta t, t_{N+1})$ ,  $x_N(t)$  is also well-defined for all  $t \in [t_N + \Delta t, t_{N+1})$ . Moreover, due to the definition of  $\pi_{N,\text{safe}}$ , Lemma B.3 implies that, with probability at least  $1 - \delta$ ,
+
+$$
+\frac {\partial h}{\partial x} \left(x _ {N}\right) \left(f \left(x _ {N}\right) + g \left(x _ {N}\right) \pi_ {N, \text {s a f e}} \left(x _ {N}\right)\right) \geq - \alpha \left(h \left(x _ {N}\right)\right) + \frac {\epsilon}{2}
+$$
+
+holds for all  $N$  and all  $x_{N}\in \mathcal{X}_{N}$ . By Lemma B.4,  $x_{N}(t)\in \mathcal{C}$  holds for all  $t\in [t_N + \Delta t,t_{N + 1})$  and all  $N\in \mathbb{N}$  with probability at least  $1 - \delta$ .
+
+(ii) The following holds with probability at least  $1 - \delta$ : For all  $x_{N}(t)$  and all  $N \in \mathbb{N}$ , if  $\alpha(h(x_{N}(t')) < \frac{\epsilon}{2}$  for any  $t' \in [t_{N} + \Delta t, t_{N+1})$ , then, by Lemma B.3 and the definition of  $\pi_{\mathrm{safe}, N}$ :
+
+$$
+\dot {h} \left(x, \pi_ {\text {s a f e}, N} \left(x _ {N} \left(t ^ {\prime}\right)\right)\right) \geq - \alpha \left(h \left(x \left(t ^ {\prime}\right)\right)\right) + \frac {\epsilon}{2} > - \frac {\epsilon}{2} + \frac {\epsilon}{2} = 0
+$$
+
+Due to continuity with respect to time, this means there exists an instance in time  $t_c \in (t', t_{N+1})$  and an  $\tilde{\varepsilon} > 0$ , such that  $\dot{h}(x_N(t), \pi_{\mathrm{saf}, N}(x_N(t))) > \tilde{\varepsilon}$  holds for all  $t \in (t', t_c]$ . By the Comparison Lemma (Khalil, 1996),  $h(x_N(t)) > h(x_N(t')) + \tilde{\varepsilon}(t_c - t') > h(x_N(t'))$  for all  $t \in (t', t_c]$ . By the same argument,  $h(x_N(t))$  is strictly increasing in  $t$  whenever  $h(x_N(t')) < h(x_N(t)) < \frac{\epsilon}{2}$ , hence  $h(x_N(t)) > h(x_N(t'))$  for all  $t \in (t_c, t_N]$ . Since  $\alpha$  is strictly increasing, this implies the desired result.
+
+(iii) The proof is identical to that of (ii), except that the strict inequality is replaced by a non-strict one.
+
+A consequence of Lemma B.5 is that safety is guaranteed if infeasibility never occurs within the safe set:
+
+Corollary B.6. Let Assumptions 2.3, 2.4 and 2.6 to 2.8 hold. With probability  $1 - \delta$ , if
+
+$$
+\max  _ {u \in \mathcal {U}} L C B _ {N} (x, u) > - \alpha (h (x)) + \frac {\epsilon}{2}, \quad \forall x \in \mathcal {C} \tag {26}
+$$
+
+holds for any  $N\in \mathbb{N}$ $\pi_{safe,N}(x)$  is well-defined for all  $x\in \mathcal{C}$  . Furthermore,  $t_{N + 1} = \infty$  and the closed-loop system
+
+$$
+\dot {x} _ {N} = f \left(x _ {N}\right) + g \left(x _ {N}\right) \pi_ {N, s a f e} \left(x _ {N}\right), \quad t \in \left[ t _ {N} + \Delta N, \infty\right),
+$$
+
+is safe.
+
+Proof. Due to the definition of  $\pi_{\mathrm{safe},N}$ , (26) directly implies that its feasible region is non-empty, i.e.,  $\pi_{\mathrm{safe},N}$  is well-defined, for all  $x \in \mathcal{C}$ . We now show that  $t_{N+1} = \infty$  by contradiction. Hence, suppose  $t_{N+1} < \infty$  holds. By Item (ii) in Lemma B.5, the trajectory  $x_N(t)$  stays strictly in the interior of  $\mathcal{C}$  for  $t \in [t_N + \Delta N, t_{N+1})$ . Due to continuity with respect to time,  $x_N(t_{N+1}) = \lim_{t \to t_{N+1}} x_N(t) \in \mathcal{C}$ . Recall that  $t_{N+1}$  denotes the earliest instance when the CBF-SOCP specifying  $\pi_{\mathrm{safe},N}$  is no longer strictly feasible. Since  $x_N(t) \in \mathcal{C}$  and  $\pi_{\mathrm{safe},N}$  is strictly feasible in  $\mathcal{C}$ , this is a contradiction, meaning  $t_{N+1} = \infty$ .
+
+# Recovering Feasibility Through Bandit Exploration
+
+We now analyze the effect of data collected under Algorithm 1 on the feasibility of the safety filter. Specifically, we demonstrate that by gathering enough data before exiting the safe set  $\mathcal{C}$ , strict feasibility is ensured anywhere in  $\mathcal{C}$ . To
+
+achieve this, we will need the following preliminary results, which bound the growth of the cumulative worst-case model error.
+
+Lemma B.7. The posterior variance satisfies
+
+$$
+\sum_ {q = 1} ^ {N} \sigma_ {i, q} ^ {2} \left(\mathbf {z} ^ {(q - 1)}\right) \leq \frac {2}{\log \left(1 + \sigma_ {i , n s} ^ {- 2}\right)} \gamma_ {i, N} \tag {27}
+$$
+
+for all  $N$ , all  $i$  and all  $\mathbf{z}^{(1)}, \ldots, \mathbf{z}^{(N)} \in \tilde{\mathcal{X}}$ .
+
+Proof. Employing the same argument as in the proof of Srinivas et al. (2012, Lemma 5.4), we get
+
+$$
+\sigma_ {i, q} ^ {2} \left(\mathbf {z} ^ {(q)}\right) \leq \frac {1}{\log \left(1 + \sigma_ {i , \mathrm {n s}} ^ {- 2}\right)} \log \left(1 + \sigma_ {i, \mathrm {n s}} ^ {- 2} \sigma_ {i, q} ^ {2} \left(\mathbf {z} ^ {(q)}\right)\right).
+$$
+
+The result then follows from Srinivas et al. (2012, Lemma 5.3).
+
+Lemma B.8. For all  $N$ , the posterior covariance matrices satisfy
+
+$$
+\sum_ {q = 1} ^ {N} t r \left(\boldsymbol {\Sigma} _ {N} \left(x ^ {(N)}, u ^ {(N)}\right)\right) \leq \sum_ {i = 1} ^ {n} \frac {2}{\log \left(1 + \sigma_ {i , n s} ^ {- 2}\right)} \gamma_ {i, N}. \tag {28}
+$$
+
+Proof. By Lemma B.7,
+
+$$
+\sum_ {q = 1} ^ {N} \operatorname {t r} \left(\boldsymbol {\Sigma} _ {N} (x ^ {(N)}, u ^ {(N)})\right) = \sum_ {q = 1} ^ {N} \sum_ {i = 1} ^ {n} \sigma_ {i, N} ^ {2} \left(\mathbf {z} ^ {(N - 1)}\right) \leq \sum_ {i = 1} ^ {n} \frac {2}{\log \left(1 + \sigma_ {i , \mathrm {n s}} ^ {- 2}\right)} \gamma_ {i, N}.
+$$
+
+Lemma B.8 bounds the growth of the posterior covariance matrices.
+
+We now show that strict feasibility in  $\mathcal{C}$  is recovered after collecting at most  $\Delta N_{\mathrm{max}}$  data points.
+
+Lemma B.9. Let Assumptions 2.3, 2.4 and 2.6 to 2.8 hold. The following holds with probability at least  $1 - \delta$ : If Algorithm 1 collects  $\Delta N_{max} - 1$  measurements in  $\mathcal{C}$ , where  $\Delta N_{max}$  satisfies
+
+$$
+\Delta N _ {m a x} > \frac {3 2 \beta_ {\Delta N _ {m a x}} ^ {2} L _ {h} ^ {2}}{\epsilon^ {2} \log (1 + \sigma_ {i , n s} ^ {- 2})} \sum_ {i = 1} ^ {n} \gamma_ {i, \Delta N _ {m a x}}
+$$
+
+then the safety filter is strictly feasible everywhere in  $\mathcal{C}$ .
+
+Proof. We employ a proof by contradiction. Hence, assume that after  $\Delta N_{\mathrm{max}} - 1$  observations have been collected, there exists an  $x^{\mathrm{c}} \in \mathcal{C}$ , such that
+
+$$
+\max  _ {u \in \mathcal {U}} L C B _ {\Delta N _ {\max }} \left(x ^ {\mathrm {c}}, u\right) \leq - \alpha \left(h \left(x ^ {\mathrm {c}}\right)\right) + \frac {\epsilon}{2}. \tag {29}
+$$
+
+In the following, we set
+
+$$
+x ^ {(\Delta N _ {\max })} \triangleq x ^ {\mathrm {c}},
+$$
+
+$$
+u ^ {(\Delta N _ {\max })} \triangleq \operatorname * {a r g   m a x} _ {u \in \mathcal {U}} L C B _ {\Delta N _ {\max }} (x ^ {\mathrm {c}}, u).
+$$
+
+Recall that Algorithm 1 only collects observations  $x^{(N)}, u^{(N)}$  if (12) is not strictly feasible, i.e.,
+
+$$
+\begin{array}{l} L C B _ {N} \left(x ^ {(N)}, u ^ {(N)}\right) \leq \max  _ {u \in \mathcal {U}} L C B _ {N} \left(x ^ {(N)}, u\right) \\ \leq - \alpha \left(h (x ^ {(N)})\right) + \frac {\epsilon}{2} \\ \end{array}
+$$
+
+for all  $u \in \mathcal{U}$ . Furthermore, due to Lemma B.3, with probability at least  $1 - \delta$ ,
+
+$$
+\begin{array}{l} \sup  _ {u \in \mathcal {U}} \dot {h} \left(x ^ {(N)}, u\right) \leq \max  _ {u \in \mathcal {U}} U C B _ {N} \left(x ^ {(N)}, u\right) \\ = U C B _ {N} \left(x ^ {(N)}, u ^ {(N)}\right), \\ \end{array}
+$$
+
+holds for all  $u\in \mathcal{U}$  and all  $N\in \mathbb{N}$ . By employing the identity
+
+$$
+L C B _ {N} (x, u) - U C B _ {N} (x, u) = - 2 L _ {h} \beta_ {N} \sqrt {\mathrm {t r} (\pmb {\Sigma} _ {N} (x , u))},
+$$
+
+we then obtain, with probability at least  $1 - \delta$ , for all  $N \in \mathbb{N}$ :
+
+$$
+\begin{array}{l} - \alpha (h (x ^ {(N)})) + \frac {\epsilon}{2} \geq L C B _ {N} \left(x ^ {(N)}, u ^ {(N)}\right) \\ \geq L C B _ {N} \left(x ^ {(N)}, u ^ {(N)}\right) + \sup  _ {u \in \mathcal {U}} \dot {h} \left(x ^ {(N)}, u\right) - U C B _ {N} \left(x ^ {(N)}, u ^ {(N)}\right) \\ = \sup  _ {u \in \mathcal {U}} \dot {h} \left(x ^ {(N)}, u\right) - 2 L _ {h} \beta_ {N} \sqrt {\operatorname {t r} \left(\Sigma_ {N} \left(x ^ {(N)} , u ^ {(N)}\right)\right)} \\ \geq - \alpha (h (x ^ {(N)})) + \epsilon - 2 L _ {h} \beta_ {N} \sqrt {\operatorname {t r} \left(\boldsymbol {\Sigma} _ {N} (x ^ {(N)} , u ^ {(N)})\right)}, \\ \end{array}
+$$
+
+where the last inequality is due to Assumption 2.6. Hence, with probability at least  $1 - \delta$ ,
+
+$$
+2 L _ {h} \beta_ {N} \sqrt {\mathrm {t r} \left(\pmb {\Sigma} _ {N} \left(x ^ {(N)} , u ^ {(N)}\right)\right)} \geq \frac {\epsilon}{2}, \quad \forall N \in \mathbb {N}.
+$$
+
+By squaring and summing up from  $N = 1$  to  $\Delta N_{\mathrm{max}}$ , then employing Lemma B.8 and (29), we get
+
+$$
+\begin{array}{l} \Delta N _ {\max } \left(\frac {\epsilon}{2}\right) ^ {2} = \sum_ {N = 1} ^ {\Delta N _ {\max }} \left(\frac {\epsilon}{2}\right) ^ {2} \\ \leq \sum_ {N = 1} ^ {\Delta N _ {\max }} \left(2 L _ {h} \beta_ {N} \sqrt {\operatorname {t r} \left(\boldsymbol {\Sigma} _ {N} \left(x ^ {(N)} , u ^ {(N)}\right)\right)}\right) ^ {2} \\ \leq \beta_ {\Delta N _ {\max }} ^ {2} 4 L _ {h} ^ {2} \sum_ {N = N _ {0}} ^ {\Delta N _ {\max }} \operatorname {t r} \left(\boldsymbol {\Sigma} _ {N} (x ^ {(N)}, u ^ {(N)})\right) \\ \leq \beta_ {\Delta N _ {\max }} ^ {2} 4 L _ {h} ^ {2} \sum_ {i = 1} ^ {n} \frac {2}{\log \left(1 + \sigma_ {i , \mathrm {n s}} ^ {- 2}\right)} \gamma_ {i, \Delta N _ {\max }}. \\ \end{array}
+$$
+
+This implies
+
+$$
+\Delta N _ {\max } \leq \frac {3 2 \beta_ {\Delta N _ {\max }} ^ {2} L _ {h} ^ {2}}{\epsilon^ {2} \log \left(1 + \sigma_ {i , \mathrm {n s}} ^ {- 2}\right)} \sum_ {i = 1} ^ {n} \gamma_ {i, \Delta N _ {\max }},
+$$
+
+which is a contradiction. Hence, the safety filter is strictly feasible in all of  $\mathcal{C}$ .
+
+Next, we analyze the amount of data collected under Algorithm 1 before potentially exiting the safe set  $\mathcal{C}$ . Specifically, we show that the closed-loop system under Algorithm 1 collects at least  $\Delta N_{\mathrm{max}}$  data points before exiting the safe set. To do so,
+
+we will require the following result:
+
+Lemma B.10. Let Assumptions 2.1, 2.3, 2.4 and 2.6 to 2.8 hold. Then, with probability at least  $1 - \delta$ : consider the closed-loop system (2) and assume there exists a time interval  $[t', t'']$  with  $t' < t''$ , such that
+
+$$
+\alpha \left(h \left(x \left(t ^ {\prime}\right)\right) \leq \frac {\epsilon}{2}, \right. \tag {30}
+$$
+
+$$
+\alpha (h (x (t)) <   \alpha (h (x (t ^ {\prime})), \quad \forall t \in (t ^ {\prime}, t ^ {\prime \prime} ],
+$$
+
+and let
+
+$$
+t _ {N ^ {\prime}} = \max  _ {N \in \mathbb {N}} t _ {N}, \quad s. t. \quad t _ {N} \leq t ^ {\prime} \tag {31}
+$$
+
+denote the last time instance when data is collected up until  $t'$ . Then, the safety filter is not strictly feasible at times  $t'$  and  $t_N$ , and
+
+$$
+| \alpha (h (x (t _ {N}))) - \alpha (h (x (t ^ {\prime}))) | <   L _ {\alpha} L _ {h} L _ {\dot {x}} \Delta t
+$$
+
+holds, where  $\Delta t$  is the sampling frequency used by Algorithm 1.
+
+Proof. We begin by showing that the safety filter is not strictly feasible at time  $t'$  by contradiction. Hence, assume that the safety filter is strictly feasible at  $t'$ . Since  $\alpha(h(x(t))) \leq \frac{\epsilon}{2}$ , by Item (iii) in Lemma B.5,  $\alpha(h(x(t))) \geq \frac{\epsilon}{2}$  holds for some  $t \in (t', t'']$ . This contradicts (30).
+
+We now show that strict feasibility does not hold at  $t_{N'}$  by contradiction. Hence, assume strict feasibility holds at  $t_{N'}$ . Due to the definition of  $t_{N'}$  (31), a new data point is not collected until strictly after  $t'$ . This implies strict feasibility at time  $t'$ , which is a contradiction.
+
+Since the safety filter is not strictly feasible at time  $t_{N'}$ , and  $t_{N'}$  is the last time instance when data is collected before  $t'$ , from the sampling frequency  $\Delta t$ , we have
+
+$$
+\Delta t = t _ {N ^ {\prime} + 1} - t _ {N ^ {\prime}} > t ^ {\prime} - t _ {N ^ {\prime}}.
+$$
+
+The result then follows from the Lipschitz continuity of  $\alpha, x$  and  $h$ .
+
+We now show that if the closed-loop system ever exits the safe set, it must collect at least  $\Delta N_{\mathrm{max}}$  data points before doing so.
+
+Lemma B.11. Let Assumptions 2.1, 2.3, 2.4 and 2.6 to 2.8 hold and let  $x(0)$  be strictly within the safe set, such that  $\alpha(h(x(0))) \geq \epsilon$ . Then the following holds with probability at least  $1 - \delta$ : if the closed-loop system under Algorithm 1 ever exits the safe set  $\mathcal{C}$ , then it collects at least  $\Delta N_{\max}$  data points before doing so.
+
+Proof. Assume that the closed-loop trajectory  $x(t)$  reaches the boundary of the safe set  $\mathcal{C}$  at some time  $t_{\mathrm{exit}}$  before exiting it:
+
+$$
+t _ {\text {e x i t}} = \sup  \left\{\hat {t} \geq 0 \mid \alpha (h (x (t)) \geq 0 \forall t \in [ 0, \hat {t} ] \right\}. \tag {32}
+$$
+
+Due to the continuity of  $x$ ,  $\alpha$  and  $h$ , and  $\alpha(h(x(0))) \geq \epsilon$ , there exists a time  $t_{\frac{\epsilon}{2}} < t_{\mathrm{exit}}$ , such that
+
+$$
+t _ {\frac {\epsilon}{2}} = \sup  \left\{\hat {t} \geq 0 \mid \alpha (h (x (\hat {t})) = \frac {\epsilon}{2}, \quad \alpha (h (x (t)) <   \frac {\epsilon}{2}, \quad \forall t \in (\hat {t}, t _ {\text {e x i t}} ] \right\}. \tag {33}
+$$
+
+Let  $N_{\frac{\epsilon}{2}}$  and  $N_{\mathrm{exit}}$  denote the number of collected observations at time  $t_{\frac{\epsilon}{2}}$  and  $t_{\mathrm{exit}}$ , respectively:
+
+$$
+N _ {\frac {\epsilon}{2}} = \arg \max  _ {N \in \mathbb {N}} t _ {N}, \quad \text {s . t .} \quad t _ {N} \leq t _ {\frac {\epsilon}{2}}
+$$
+
+$$
+N _ {\text {e x i t}} = \arg \max  _ {N \in \mathbb {N}} t _ {N}, \quad \text {s . t .} \quad t _ {N} \leq t _ {\text {e x i t}}.
+$$
+
+Then, with probability at least  $1 - \delta$ : Due to Lemma B.10, we have
+
+$$
+\left| \alpha \left(h \left(x \left(t _ {N _ {\frac {\epsilon}{2}}}\right)\right)\right) - \alpha \left(h \left(x \left(t _ {\frac {\epsilon}{2}}\right)\right)\right) \right| \leq L _ {\alpha} L _ {h} L _ {\dot {x}} \Delta t \tag {34}
+$$
+
+$$
+\left| \alpha \left(h \left(x \left(t _ {N _ {\mathrm {e x i t}}}\right)\right)\right) - \alpha \left(h \left(x \left(t _ {\mathrm {e x i t}}\right)\right)\right) \right| \leq L _ {\alpha} L _ {h} L _ {\dot {x}} \Delta t. \tag {35}
+$$
+
+We now analyze the growth of  $\alpha(h(x(t)))$  between  $t = t_N$  and  $t = t_{N+1}$ . We separately analyze the case where the safety filter is strictly feasible and the case where it is not. If the safety filter is not strictly feasible at time  $t_{N-1}$ , due to Assumption 2.1 and the sampling frequency  $\Delta t$ , the CBF changes between  $t_{N-1}$  and  $t_N$  at most by
+
+$$
+\begin{array}{l} \left| \alpha (h (x (t _ {N}))) - \alpha (h (x (t _ {N - 1}))) \right| \\ \leq L _ {\dot {x}} L _ {\alpha} L _ {h} \left| t _ {N} - t _ {N - 1} \right| \tag {36} \\ \leq L _ {\dot {x}} L _ {\alpha} L _ {h} \Delta t. \\ \end{array}
+$$
+
+By Item (iii) in Lemma B.5, if the safety filter is strictly feasible at time  $t_{N-1}$  and  $\alpha(h(x(t_{N-1}))) \leq \frac{\epsilon}{2}$ , then  $\alpha(h(x(t_{N}))) \geq \alpha(h(x(t_{N-1}))$ . Hence, for all  $t_{N-1} \in [t_{\frac{\epsilon}{2}}, t_{\mathrm{exit}}]$ , whether the safety filter is feasible or not,
+
+$$
+\alpha \left( \right.h \left(x \left(t _ {N - 1}\right)\right) - \alpha \left(h \left(x \left(t _ {N}\right)\right)\right) \leq L _ {\dot {x}} L _ {\alpha} L _ {h} \Delta t \tag {37}
+$$
+
+holds. We then have
+
+$$
+\begin{array}{l} \frac {\epsilon}{2} = \alpha \left(h \left(x \left(t _ {\frac {\epsilon}{2}}\right)\right) - \alpha \left(h \left(x \left(t _ {\text {e x i t}}\right)\right) \right. \right. \\ = \alpha (h (x (t _ {\frac {\epsilon}{2}})) - \alpha (h (x (t _ {\mathrm {e x i t}})) + \alpha (h (x (t _ {N _ {\mathrm {e x i t}}})) - \alpha (h (x (t _ {N _ {\frac {\epsilon}{2}}})) - \alpha (h (x (t _ {N _ {\mathrm {e x i t}}})) + \alpha (h (x (t _ {N _ {\frac {\epsilon}{2}}})) \\ \leq \alpha (h (x (t _ {N _ {\frac {\epsilon}{2}}})) - \alpha (h (x (t _ {N _ {\mathrm {e x i t}}})) + 2 L _ {\alpha} L _ {h} L _ {\dot {x}} \Delta t \\ = \sum_ {N = N _ {\frac {\epsilon}{2}} + 1} ^ {N _ {\text {e x i t}}} \alpha (h (x (t _ {N - 1}))) - \alpha (h (x (t _ {N}))) + 2 L _ {\alpha} L _ {h} L _ {\dot {x}} \Delta t \\ \leq \left(N _ {\text {e x i t}} - N _ {\frac {2}{3}} - 1\right) L _ {\alpha} L _ {h} L _ {\dot {x}} \Delta t + 2 L _ {\alpha} L _ {h} L _ {\dot {x}} \Delta t \\ = \left(N _ {\text {e x i t}} + 1 - N _ {\frac {\epsilon}{2}}\right) L _ {\alpha} L _ {h} L _ {\dot {x}} \Delta t, \\ \end{array}
+$$
+
+where the first inequality is due to (34), and the second inequality is due to (37). This implies
+
+$$
+N _ {\text {e x i t}} - N _ {\frac {\epsilon}{2}} + 1 \geq \frac {\epsilon}{2 L _ {\alpha} L _ {h} L _ {\dot {x}} \Delta t} > \Delta N _ {\max },
+$$
+
+where the last equality follows from the sampling frequency specification (16). Since  $N_{\mathrm{exit}}$ ,  $N_{\frac{\varepsilon}{2}}$  and  $\Delta N_{\max}$  are integers, this implies that the amount of collected data satisfies  $N_{\mathrm{exit}} - N_{\frac{\varepsilon}{2}} \geq \Delta N_{\max}$ .
+
+We now put everything together and prove Theorem 3.2.
+
+Proof of Theorem 3.2. By Lemma B.11, Algorithm 1 collects at least  $\Delta N_{\mathrm{max}}$  observations before ever exiting the safe set  $\mathcal{C}$ . Lemma B.9 states that the safety filter is strictly feasible within all of  $\mathcal{C}$  after collecting at most  $\Delta N_{\mathrm{max}}$  observations. Together with Corollary B.6 this implies that, with probability at least  $1 - \delta$ , the closed-loop system is safe and Algorithm 1 collects at most  $\Delta N_{\mathrm{max}}$  observations.
+
+# C. Kernel-Dependent Bounds for  $\Delta N_{\mathrm{max}}$
+
+Theorem 3.2 specifies that
+
+$$
+\Delta N _ {\max } > \frac {3 2 \beta_ {\Delta N _ {\max }} ^ {2} L _ {h} ^ {2}}{\epsilon^ {2} \log \left(1 + \sigma_ {i , \mathrm {n s}} ^ {- 2}\right)} \sum_ {i = 1} ^ {n} \gamma_ {i, \Delta N _ {\max }} \tag {38}
+$$
+
+is a sufficient condition to guarantee safety. We now analyze how the right-hand side term behaves for different kernels. We begin by analyzing how the maximal information gain  $\gamma_{i,N}$  depends on the maximal information gain of the kernels  $k_{f_i}$  and  $k_{g_{i,j}}$  used to model  $f$  and  $g$ .
+
+Lemma C.1. Let  $k_{i}$  be given as in (6), and let  $\gamma_{f_i,N}$  and  $\gamma_{g_{i,j},N}$  denote the maximal information gain after  $N$  rounds of data collection under the kernels  $k_{f_i}$  and  $k_{g_{i,j}}$ , respectively. Then
+
+$$
+\gamma_ {i, N} \leq \gamma_ {f _ {i}, N} + \sum_ {j = 1} ^ {m} \gamma_ {g _ {i, j}, N} + 3 m \log (N) \tag {39}
+$$
+
+Proof. The proof follows directly the definition of the composite kernel  $k_{i}$ , Krause & Ong (2011, Theorem 2) and Krause & Ong (2011, Theorem 3).
+
+We now dissect the right-hand side of (40). Define  $\gamma_{\Delta N_{\mathrm{max}}} \triangleq \max_{i \leq n} \gamma_{i, \Delta N_{\mathrm{max}}}$  and  $B \triangleq \max_{i \leq n} B_i$ . From the definition of  $\beta_{\Delta N_{\mathrm{max}}}$ , it follows that
+
+$$
+\begin{array}{l} \beta_ {\Delta N _ {\max }} ^ {2} \sum_ {i = 1} ^ {n} \gamma_ {i, \Delta N _ {\max }} \leq \beta_ {\Delta N _ {\max }} ^ {2} n \gamma_ {\Delta N _ {\max }} \\ = n \gamma_ {\Delta N _ {\max }} B ^ {2} + n \gamma_ {\Delta N _ {\max }} 2 B \sigma_ {i, \mathrm {n s}} \sqrt {2 \left(\gamma_ {\Delta N _ {\max }} + 1 + \log \left(n \delta^ {- 1}\right)\right)} + n \gamma_ {\Delta N _ {\max }} \sigma_ {i, \mathrm {n s}} ^ {2} 2 \left(\gamma_ {\Delta N _ {\max }} + 1 + \log \left(n \delta^ {- 1}\right)\right) \\ \leq C _ {1} (B, \sigma_ {i, \mathrm {n s}}, n) \gamma_ {\Delta N _ {\max}} + C _ {\frac {3}{2}} (B, \sigma_ {i, \mathrm {n s}}, n) \gamma_ {\Delta N _ {\max}} ^ {\frac {3}{2}} + C _ {2} (B, \sigma_ {i, \mathrm {n s}}, n) \gamma_ {\Delta N _ {\max}} ^ {2} \\ \leq \left(C _ {1} (B, \sigma_ {i, \mathrm {n s}}, n) + C _ {\frac {3}{2}} (B, \sigma_ {i, \mathrm {n s}}, n) + C _ {2} (B, \sigma_ {i, \mathrm {n s}}, n)\right) \max  \left\{\gamma_ {\Delta N _ {\max }} ^ {2}, 1 \right\} \\ \leq \left(C _ {1} (B, \sigma_ {i, \mathrm {n s}}, n) + C _ {\frac {3}{2}} (B, \sigma_ {i, \mathrm {n s}}, n) + C _ {2} (B, \sigma_ {i, \mathrm {n s}}, n)\right) \left(\gamma_ {\Delta N _ {\max }} ^ {2} + 1\right), \\ \end{array}
+$$
+
+where
+
+$$
+C _ {1} (B, \sigma_ {i, \mathrm {n s}}, n) = n B ^ {2} + n 2 B \sigma_ {i, \mathrm {n s}} \sqrt {2 \left(1 + \log \left(n \delta^ {- 1}\right)\right)} + n \sigma_ {i, \mathrm {n s}} ^ {2} 2 \left(1 + \log \left(n \delta^ {- 1}\right)\right)
+$$
+
+$$
+C _ {\frac {3}{2}} (B, \sigma_ {i, \mathrm {n s}}, n) = n 2 B \sigma_ {i, \mathrm {n s}} \sqrt {2}
+$$
+
+$$
+C _ {2} (B, \sigma_ {i, \mathrm {n s}}, n) = 2 n \sigma_ {i, \mathrm {n s}} ^ {2}.
+$$
+
+For most commonly used kernels, the growth rate of the maximal information gain is of the form  $\mathcal{O}(N^{\omega}\log (N))$ , where  $0\leq \omega < 1$  (Srinivas et al., 2012). Hence, using Lemma C.1, we can bound  $\gamma_{N}$  as  $\gamma_N\leq C_\gamma N^\omega (\log (N))^{\theta}$  for some positive  $C_\gamma$ ,  $\omega$  and  $\theta$ , and  $N$  large enough. Using an estimate of this type, we can compute a (potentially crude) upper bound for the right-hand side of (40) as
+
+$$
+\frac {3 2 \beta_ {\Delta N _ {\max}} ^ {2} L _ {h} ^ {2}}{\epsilon^ {2} \log (1 + \sigma_ {i , \mathrm {n s}} ^ {- 2})} \sum_ {i = 1} ^ {n} \gamma_ {i, \Delta N _ {\max}} \leq C (B, \sigma_ {i, \mathrm {n s}}, n) \left(C _ {\gamma} \Delta N _ {\max } ^ {2 \omega} (\log (N)) ^ {\theta} + 1\right) \leq \tilde {C} (B, \sigma_ {i, \mathrm {n s}}, n) \Delta N _ {\max } ^ {2 \omega} (\log (N)) ^ {2 \theta} \tag {40}
+$$
+
+where  $C$  and  $\tilde{C}$  are appropriate constants. This yields the sufficient condition for choosing  $\Delta N_{\mathrm{max}}$
+
+$$
+\Delta N _ {\max } \geq \left(\tilde {C} (B, \sigma_ {i, \mathrm {n s}}, n) (\log (\Delta N _ {\max }) ^ {2 \theta}\right) ^ {\frac {1}{1 - 2 \omega}}, \tag {41}
+$$
+
+which can easily be solved using numerical methods. For squared-exponential kernels,  $\omega = 0$  and  $\theta = n$ . For linear kernels,  $\omega = 0$  and  $\theta = 1$ , whereas for Matérn kernels,  $\theta = 0$  and  $\omega$  depends on its smoothness parameter (Srinivas et al., 2012).

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+# Learning Safety Constraints for Large Language Models
+
+Xin Chen<sup>1</sup> Yarden As<sup>1</sup> Andreas Krause<sup>1</sup>
+
+# Abstract
+
+Large language models (LLMs) have emerged as powerful tools but pose significant safety risks through harmful outputs and vulnerability to adversarial attacks. We propose SaP—short for Safety Polytope—a geometric approach to LLM safety, that learns and enforces multiple safety constraints directly in the model's representation space. We develop a framework that identifies safe and unsafe regions via the polytope's facets, enabling both detection and correction of unsafe outputs through geometric steering. Unlike existing approaches that modify model weights, SaP operates post-hoc in the representation space, preserving model capabilities while enforcing safety constraints. Experiments across multiple LLMs demonstrate that our method can effectively detect unethical inputs, reduce adversarial attack success rates while maintaining performance on standard tasks, thus highlighting the importance of having an explicit geometric model for safety. Analysis of the learned polytope facets reveals emergence of specialization in detecting different semantic notions of safety, providing interpretable insights into how safety is captured in LLMs' representation space.
+
+# 1. Introduction
+
+Large Language Models (LLMs) have demonstrated remarkable capabilities across diverse tasks, yet their increasing real-world deployment raises urgent safety concerns—these models can generate harmful content or be manipulated through adversarial attacks. While approaches like Reinforcement Learning from Human Feedback (RLHF) show promise, they face fundamental limitations. For instance, models can learn to "game" reward functions rather than become genuinely safer, and noisy specification of human preferences leads to unintended behaviors (Casper et al., 2023).
+
+$^{1}$ Department of Computer Science, ETH Zürich, Zürich, Switzerland. Correspondence to: Xin Chen <xin.chen@inf.ethz.ch>.
+
+Proceedings of the  $42^{nd}$  International Conference on Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).
+
+Current approaches to LLM safety span a broad spectrum yet face significant tradeoffs. Prompt-based techniques attempt to alter model behavior through input/output engineering; however, they prove brittle and easily circumvented (Kandpal et al., 2023). Training-time approaches that rely on safe RLHF (Dai et al., 2023; Wachi et al., 2024; Rame et al., 2024) aim to balance helpfulness with safety but require expensive data relabeling and model retraining, lacking post-hoc mechanisms to ensure safety. In addition, it is often difficult to interpret how these methods make safety-related decisions. For instance, it is unclear how these methods quantify the severity of unsafe users' requests, or how they pinpoint the underlying cause that makes a request unsafe. This landscape suggests the need for the development of methods that leverage inference-time techniques while offering intuitive means for interpretability.
+
+Motivated by recent work on constraint learning from demonstrations in constrained Markov decision processes (CMDP, Altman, 1999; Lindner et al., 2024), our key insight is that LLM safety can be framed as a geometric constraint learning problem. This geometric perspective allows us to model an explicit safe set, defined as a polytope, within the model's representation space. Each of the polytope's facets represents a different constraint that must be satisfied. As we later show, while learning these constraints solely depends on binary safe/unsafe labels, our approach associates these constraints with different intuitive semantic meanings of safety.
+
+We develop SaP to realize this geometric approach. Our approach comprises two key components: a concept encoder that disentangles different safety concepts, and a steering algorithm that guides unsafe outputs back into the safe set while preserving model capabilities. Our framework and its components are illustrated in Figure 1 to provide a clearer understanding of our approach.
+
+# Contributions.
+
+- We propose a novel geometric framework that yields an explicit model of safety in LLMs, formulated as a polytope in models' representation space.  
+- An inference-time steering algorithm that guides unsafe outputs back to the safe set while preserving model capabilities, providing defense against state-of-the-art adversarial attacks.
+
+![](images/49287b31b79b2b9944cd7a5e9bdc9d441d8b724f7729fc22e87894dd9e7268c2.jpg)  
+Figure 1: Illustration of the geometric approach to language model safety proposed in this paper. A safety facet is triggered when a user asks the model to write a response suggesting fabricated content. When the model generates responses, we steer its internal representation back to the safe region to produce safe outputs.
+
+- An extensive empirical analysis of learned polytope facets, which reveals naturally emerging specialization in detecting different semantic concepts of safety.
+
+Our code is publicly available at https://github.com/lasgroup/SafetyPolytope.
+
+# 2. LLM Safety through the Lens of Constrained Markov Decision Processes
+
+We tackle LLM safety as a sequential decision-making problem, where models must reason about long-term consequences of their decisions, i.e., their choice of words. An intuitive approach for modeling safety requirements is by imposing some constraints within this decision-making process. Constraint Markov decision processes pose a natural formulation for such problems, allowing harmful behaviors to be modeled as constraints.
+
+Constrained Markov decision processes. We define an infinite-horizon discounted constrained Markov decision processes (Altman, 1999, CMDP) as the tuple  $(\mathcal{X},\mathcal{A},r,\{c_j\}_{j = 1}^n,p,\gamma ,\rho)$ , where  $\mathcal{X}$  and  $\mathcal{A}$  are state and action spaces respectively,  $p(\pmb{x}'\mid \pmb{x},\pmb{a})$ , describes a probability distribution over the next state  $\pmb{x}'$ , given a state  $\pmb{x}\in \mathcal{X}$  and an action  $\pmb{a}\in \mathcal{A}$ . States are initially drawn from  $\rho (\pmb{x}_0)$ , the initial state distribution. A reward function  $r(\pmb{x},\pmb{a})$  is to be maximized, whereas  $c_{j}(\pmb {x},\pmb {a}),j\in \{1,\ldots ,n\}$ , are cost functions that must remain bounded. The goal is to find a stationary policy  $\pi (\pmb {x}\mid \pmb {a})$  that solves  $\max_{\pi \in \mathcal{F}}\mathbb{E}_{\pi}[\sum_{t = 0}^{\infty}\gamma^{t}r(\pmb{x}_{t},\pmb{a}_{t})]$  where  $\mathcal{F}:= \{\pi |\forall j\mathbb{E}_{\pi}[\sum_{t = 0}^{\infty}\gamma^{t}c_{j}(\pmb{x}_{t},\pmb{a}_{t})]\leq \xi_{j}\}$  and  $\xi_{j}$  are predefined cost budget paramters. Constraints in CMDPs can be equivalently expressed in terms of the value functions of the cost objectives. Specifically, let
+
+$v_{j}^{\pi}(\pmb {x}) = \mathbb{E}_{\pi}\left[\sum_{t = 0}^{\infty}\gamma^{t}c_{j}(\pmb{x}_{t},\pmb{a}_{t})\mid \pmb{x}_{0} = \pmb {x}\right]$  denote the cost value function for cost  $c_{j}$ . Then, the constraint can be rewritten as  $\mathbb{E}_{\rho}\left[v_{j}(\pmb{x}_{0})\right]\leq \xi_{j}$ .
+
+Natural language as a Markov decision process. In the case of language, one can think of a token-level MDP. In this formulation, a finite set of tokens  $\mathcal{V}$  represents the language's vocabulary, the state space is a set of all (possibly infinitely long) sequences  $\mathcal{X} = \bigcup_{H=1}^{\infty} \mathcal{V}^{H}$ , where  $\mathcal{V}^{H} = \mathcal{V} \times \dots \times \mathcal{V}$  ( $H$  times), an  $H$ -fold Cartesian product of all tokens in  $\mathcal{V}$ . Indeed, LLMs have proven highly effective in meaningfully "navigating" this combinatorially vast state space. Models predict only the next token, hence the action space is  $\mathcal{A} = \mathcal{V}$ . The vocabulary includes two special tokens ' <eos> and ' <bos >', that represent the MDP's terminal and initial states respectively. The initial state distribution  $\rho(\boldsymbol{x}_0) = \mathbb{1}_{\boldsymbol{x}_0 \in \{\langle \text{bos} \rangle\}}$ , hence trajectories always start from the token ' <bos >'. For an action  $\boldsymbol{a} \in \mathcal{V}$ , the next state is  $\boldsymbol{x}' = \boldsymbol{x} \| \boldsymbol{a}$  where the operation  $(\cdot \| \cdot)$  represents a concatenation. A special transition is reserved for when  $\boldsymbol{a} = <eos>$ , leading to a terminal state, where  $\boldsymbol{x}'$  is fixed to  $\boldsymbol{x}$  forever. An autoregressive language model can be seen as a policy  $\pi(\boldsymbol{a} | \boldsymbol{x})$  for determining the next token given all the previous tokens. While using MDPs to describe autoregressive models and language has been extensively studied in several previous works (Cundy & Ermon, 2023; Rafailov et al., 2024; Zeng et al., 2024), the main argument of this work is that safe and ethical language not only tries to maximize a reward function, but is also subject to some constraints. For instance, LLMs should avoid using language that incites harm or solicits unlawful actions. We argue that these constraints are intrinsically reflected in human natural language and should be modeled to ensure model safety and control. While manually specifying such constraints is nuanced due to their complexity and context-dependence, this work in
+
+produces a framework for learning them directly from data.
+
+Learning constraints from demonstrations. Our main result relies on the idea that for CMDPs, a conservative estimate of  $\mathcal{F}$  can be learned only from trajectory demonstrations. More concretely, Lindner et al. (2024) establish that given a dataset of trajectories that were drawn from  $k$  safe policies  $\pi_1,\ldots ,\pi_k$ , and a feature representation  $\mathbf{f}:\mathcal{X}\times \mathcal{A}\to [0,1]^d$  such that  $\mathbf{f}(\pi)\coloneqq \mathbb{E}_{\pi}[\sum_{t = 0}^{\infty}\gamma^{t}\mathbf{f}(\pmb{x}_{t},\pmb{a}_{t})]$ , one can construct a conservative approximation of  $\mathcal{F}$  with any convex combination of the feature expectations of  $\pi_1,\dots ,\pi_k$
+
+$$
+\mathcal {Q} := \left\{\pi \left| \sum_ {i = 1} ^ {k} \lambda_ {i} \mathbf {f} \left(\pi_ {k}\right), \lambda_ {i} \geq 0, \sum_ {i = 1} ^ {k} \lambda_ {i} = 1 \right. \right\}.
+$$
+
+Crucially, the set  $\mathcal{Q}$  is a convex polyhedron, i.e., a set of linear equations that determines the feasibility of policies. Furthermore, Lindner et al. (2024) show that safety constraints can be learned as linear mappings of feature expectations  $\mathbf{f}$ , independently from rewards.
+
+The geometry of constraints in LLMs. The above two observations—that  $\mathcal{Q}$  is a convex polyhedron, and that constraints are linear w.r.t. feature expectations— motivate a geometric interpretation for safety in LLMs. We propose viewing safety constraints through a geometric lens, representing them via a polytope—the intersection of  $k$  halfspaces defined by linear inequalities  $\tilde{\mathcal{Q}} := \left\{\tilde{\mathbf{f}} \mid \boldsymbol{\phi}^{\top} \tilde{\mathbf{f}} \leq \tilde{\boldsymbol{\xi}}\right\}$ . Here,  $\boldsymbol{\phi} \in \mathbb{R}^{d \times K}$  represents  $K$  hyperplanes in  $d$ -dimensional space,  $\tilde{\boldsymbol{\xi}} \in \mathbb{R}^{K}$  are thresholds, and  $\tilde{\mathbf{f}} \in \mathbb{R}^{d}$  is the input feature vector. The features  $\tilde{\mathbf{f}}$  are learned during the pre-training phase of LLMs. The linear representation hypothesis (Park et al., 2023; 2024) provides empirical evidence that high-level semantic concepts in LLMs naturally manifest as linear directions in the feature space of  $\tilde{\mathbf{f}}$ . This suggests that safety constraints in language, like those in CMDPs, have an inherent geometric structure, further grounding our approach of learning polytope facets in the LLM's feature space. This formulation of safety facets as geometric constraints enables post-hoc safety control without model fine-tuning, while providing a tool for analyzing the learned safety concepts.
+
+# 3. Safety Polytope (SaP)
+
+In this section, we present our approach for learning the facets from demonstration data and demonstrate the utility of our approach for interpretability and steering of unsafe output generation. On a high-level, our approach is composed of three main steps: (i) using a labeled dataset to obtain features  $\tilde{\mathbf{f}}$  from a pre-trained LLM and (ii) determining the polytope's hyperplanes' parameters  $\phi$  and  $\tilde{\xi}$ . (iii) using the learned polytope for steering unsafe outputs into the safe region.
+
+Feature extraction. Our aim is to use a pre-trained LLM to extract features  $\tilde{\mathbf{f}}$ , which can then be used to construct our polytope. Concretely, given a dataset  $\{(x^i,y^i)\}_{i = 1}^N$  consisting of token sequences  $x^{i}\in \mathcal{X}$  and their corresponding labels  $y^{i}\in \{-1, + 1\}$ , indicating whether the token sequence is considered safe. Following our conceptual motivation from the previous section, this can be thought of as a case, where for some  $j$  and  $\pi$ , the label indicates whether  $v_{j}^{\pi}(\pmb{x}^{i})\leq \xi_{j}$ . These labels are commonly derived from human annotations on aspects such as ethics, toxicity, and privacy protection (Ji et al., 2023; Gehman et al., 2020; Lin et al., 2021) or from success/failure of adversarial attacks. A simple method for feature extraction is to perform a forward pass on each example and collect the model's intermediate features (Arditi et al., 2024),  $\pmb {h}\in \mathbb{R}^{d_h}$ . Denoting the intermediate features of layer  $l$  as  $\bar{\pi}_l$ , we get  $\pmb {h}^i = \bar{\pi}_l(\pmb {x}^i)$ , yielding data  $\{(h^i,y^i)\}_{i = 1}^N$ . To determine a full sentence's safety, we use the feature  $\pmb{h}^i$  obtained at the last word of a sentence, i.e., when the last token of  $\pmb{x}^i$  is <eos>.
+
+Convex Polytope Machines. A key challenge in constraint learning for LLMs is scaling the estimation of the polytope's facets  $\phi$  to high dimensions. For instance, the QuickHull algorithm for convex-hull vertex identification (Barber et al., 1996), employed by Lindner et al. (2024), becomes prohibitively intractable due to its  $\mathcal{O}(N^{\lfloor d_h / 2\rfloor})$  time complexity. To overcome this, we adopt the CPM algorithm (Convex Polytope Machine, Kantchelian et al., 2014). Instead of explicitly computing a convex-hull, CPM treats the computation of polytopes as a classification problem. Concretely, given a dataset  $\{(h^i,y^i)\}_{i = 1}^N$ , where the labels indicate whether a point  $h^i$  lies outside  $(-1)$  or inside  $(+1)$  the polytope, and for this discussion only, suppose  $d_{h} = d$  to maintain consistent notation, CPM learns a decision boundary by minimizing
+
+$$
+\sum_ {i \in \mathcal {I} _ {+ 1}} \sum_ {k = 1} ^ {K} \left[ \kappa + \phi_ {k} ^ {\top} \boldsymbol {h} ^ {i} \right] _ {+} + \sum_ {i \in \mathcal {I} _ {- 1}} \left[ \kappa - \phi_ {z (\boldsymbol {h} ^ {i})} ^ {\top} \boldsymbol {h} ^ {i} \right] _ {+} + \lambda_ {\phi} \| \phi \| ^ {2}.
+$$
+
+Above,  $[\cdot]_{+} := \max(0, \cdot)$ ,  $\mathcal{I}_{+1} = \{i \mid y^{i} = +1\}$  and  $\mathcal{I}_{-1} = \{i \mid y^{i} = -1\}$ ,  $\kappa > 0$  is a margin parameter,  $\lambda_{\phi}$  is a regularization parameter and  $z(\cdot)$  assigns input points to their corresponding facet (see Appendix A). Kantchelian et al. (2014) provide more details on the connection of the CPM loss to Support Vector Machines. Crucially, optimizing the above loss allows us to leverage gradient descent to learn  $\phi$ , effectively mitigating scalability challenges associated with the dataset size  $N$  and feature dimension  $d_{h}$ .
+
+Untangling word ambiguity. Learning polytopes with gradient descent enables us to use tools from the interpretability literature to associate facets with intuitive, "human-friendly", concepts of safety. Concretely, model representations are often susceptible to polysemanticity (Elhage
+
+et al., 2022), a phenomenon where the same model activations are triggered by multiple distinct concepts. Drawing inspiration from sparse autoencoders (Cunningham et al., 2023), we propose three modifications to the pipeline introduced above. First, we introduce a linear layer followed by a ReLU function to provide nonlinearity on top of  $\pmb{h}$ , referred to as the 'Concept Encoder',  $\mathrm{E}_C: \mathbb{R}^{d_h} \to \mathbb{R}^d$ , producing a dataset of features and labels  $\{(\tilde{\mathbf{f}}^i = \mathrm{E}_C(\bar{\pi}(\boldsymbol{x}^i)), y^i)\}_{i=1}^N$ . Second, together with  $\phi$ , we make the safety threshold  $\xi$  a learnable parameter. Finally, we include a sparsity regularization term on  $\tilde{\mathbf{f}}$ . These modifications result in the following training loss:
+
+$$
+\begin{array}{l} L (\phi , \tilde {\xi}) := \sum_ {i \in \mathcal {I} _ {+ 1}} \sum_ {k = 1} ^ {K} \left[ \kappa + \phi_ {k} ^ {\top} \tilde {\mathbf {f}} ^ {i} - \tilde {\xi} _ {k} \right] _ {+} \\ + \sum_ {i \in \mathcal {I} _ {- 1}} \left[ \kappa - \phi_ {z (\tilde {\mathbf {f}} ^ {i})} ^ {\top} \tilde {\mathbf {f}} ^ {i} + \tilde {\xi} _ {z (\tilde {\mathbf {f}} ^ {i})} \right] _ {+} + \lambda_ {\tilde {\mathbf {f}}} \| \tilde {\mathbf {f}} ^ {i} \| _ {1} ^ {2} + \lambda_ {\phi} \| \phi \| ^ {2} \tag {1} \\ \end{array}
+$$
+
+Intuitively, the inclusion of the sparsity regularization term  $\lambda_{\tilde{\mathbf{f}}}$  induces feature sparsity, which empirically reduces polysemanticsy when assigning inputs to different concepts of safety. We highlight that, while this training objective is non-smooth due to the sparsity regularization term, we can still compute its sub-gradients and use standard SGD-based learning techniques to optimize it w.r.t.  $\phi$ . Additional implementation details are provided in Appendix A. In Section 4, we analyze the correspondence between human concepts of safety (e.g., violent language) and polytope edges and ablate the use of the additional non-linear transformation, demonstrating the critical role of these components.
+
+Representation steering. Obtaining  $\phi$  allows us to directly verify whether next-token predictions by our model lie within the polytope  $\tilde{\mathcal{Q}}$ —and are therefore safe—by evaluating  $\phi^{\top}\tilde{\mathbf{f}} \leq \tilde{\boldsymbol{\xi}}$ . An immediate implication of this insight is that, before exposing the next token to users, we can adjust activations dynamically, i.e., within the model's response generation loop. Building on methodology from constrained reinforcement learning (c.f. Dalal et al., 2018), we propose solving the following optimization problem:
+
+$$
+\min  _ {\boldsymbol {h}} \| \bar {\pi} _ {l} (\boldsymbol {x}) - \boldsymbol {h} \| _ {1} \text {s . t .} \phi^ {\top} \tilde {\mathbf {f}} (\boldsymbol {h}) \leq \tilde {\xi}. \tag {2}
+$$
+
+Importantly, as long as the model's representations already reside within  $\tilde{\mathcal{Q}}$ , the original token generation remains unchained. Algorithm 1 provides a detailed token generation loop that actively adjusts outputs to ensure safety when necessary. Note that in practice, obtaining an approximate solution Equation (2) can be achieved using first-order methods (Bertsekas, 2016). For example, in our experiments, taking only a few steps of a simple Lagrangian relaxation (see Appendix B) substantially improves safety performance.
+
+Algorithm 1 SafeFlow: Representation Steering for Safe Response Generation
+
+Require:  $\pi, x_0, \phi, \tilde{\xi}$
+
+$$
+\begin{array}{l} \boldsymbol {x} _ {t} \gets \boldsymbol {x} _ {0} \\ \text {w h i l e} x _ {t} [ \text {e n d} ] \neq <   \mathrm {e o s} > \mathbf {d o} \\ \end{array}
+$$
+
+Obtain  $\bar{\pi}_l(\pmb{x}_t)$
+
+$$
+\begin{array}{l} \boldsymbol {h} \leftarrow \arg \min  _ {\boldsymbol {h}} \| \bar {\pi} _ {l} (\boldsymbol {x}) - \boldsymbol {h} \| _ {1} \text {s . t .} \phi^ {\top} \tilde {\mathbf {f}} \leq \tilde {\xi} \quad > \text {E q .} 2 \\ \bar {\pi} _ {l} \leftarrow h \quad > \text {O v e r r i d e a c t i v a t i o n s o f l a y e r} l \\ \boldsymbol {a} _ {t} \leftarrow \pi (\boldsymbol {a} _ {t} \mid \boldsymbol {x} _ {t}) \quad > \text {C o m p l e t e p a s s , d e c o d e n e x t t o k e n} \\ \boldsymbol {x} _ {t} \leftarrow \boldsymbol {x} _ {t} \| \boldsymbol {a} _ {t} \\ \end{array}
+$$
+
+end while
+
+output  $x_{t}$
+
+Our approach scales efficiently to large batches via existing tools for vectorized computation (Bradbury et al., 2018; Agrawal et al., 2019; Blondel et al., 2022; Lu et al., 2024) and aligns with the paradigm shift towards "test time compute" (OpenAI, 2024) to improve safety in tandem with reasoning capabilities.
+
+# 4. Experiments
+
+We evaluate SaP on three LLMs: Llama2-7B (Touvron et al., 2023), Ministral-8B (Jiang et al., 2024), and Qwen2-1.5B (Yang et al., 2024). Section 4.1 presents our main results on defending against adversarial attacks using SaP's steering method in Algorithm 1. In Section 4.2, we demonstrate SaP's interpretability via the Concept Encoder and its defense performance when evaluated on the BeaverTails dataset(Ji et al., 2023). Next, Section 4.3 analyzes the impact of polytope facets through ablation studies, with discussions on model design.
+
+# 4.1. Tactful Responses via Representation Steering
+
+We evaluate the steering capability of Algorithm 1 on Harm-Bench (Mazeika et al., 2024), a standardized benchmark for assessing LLM safety against adversarial attacks. We select nine representative attack methods including gradient-based approaches: GCG (Zou et al., 2023), GBDA (Guo et al., 2021), AutoPrompt (Shin et al., 2020), PEZ (Wen et al., 2023), UAT (Wallace et al., 2021), AutoDAN (Liu et al., 2023) and human-crafted jailbreaks: Human Jailbreak (Shen et al., 2024) and Direct Request (Mazeika et al., 2024). We additionally evaluate on Adaptive Attack (Andriushchenko et al., 2024), a state-of-the-art adversarial attack method, on JailBreakBench (Chao et al., 2024).
+
+For each model, we train SaP using the following procedure: (i) we identify the top 3 most effective attack methods based on their success rates; (ii) for these methods, we collect model features from  $80\%$  of their attack strings for
+
+![](images/851d9c3df2adc40b0831c6ce95a67d8e1968df111cbaad253c94fb72d7ef2d55.jpg)  
+Figure 2: Comparison of model MMLU accuracy and average Attack Success Rate (ASR) on 9 attack algorithms. All defense methods are evaluated over 5 seeds. SaP consistently retains the original MMLU accuracy while having significantly lower ASR compared to most baseline methods.
+
+training, reserving  $20\%$  for testing; (iii) we then evaluate SaP's effectiveness against all nine attack methods. Our extensive evaluations test the trained polytope's robustness and generalization ability across both attack methods and adversarial attack benchmarks. Detailed training setup can be found in Appendix C.1.
+
+For each attack method, we compare SaP against 6 defense baselines. Our comparison includes (i) Prompt-based defense methods: Self Reminder (Xie et al., 2023), Response Check (Wang et al., 2024), and SmoothLLM (Robey et al., 2024). These baselines defend against unreasonable LLM requests by prompting, filtering, or defaulting to a rejection string when unsafe content is detected. (ii) Training-based defense methods: Rejection Sampling that rejects a request based on the polytope's binary decision. We further compare against MLP where we use an MLP model with equivalent parameter count as our SaP model. It is first trained as a safety classifier using binary cross entropy (BCE) loss, then used for the same steering procedure as SaP. We also train MLP with 5 extra layers (referred to below as  $\mathrm{MLP} + 5$ ) that has the same setup as MLP but with 5 extra layers. Along with SaP, we test all 7 defense methods on the selected 9 attack algorithms, with each setup repeated over 5 seeds.
+
+We evaluate all methods' capability on The Massive Multi-task Language Understanding benchmark (Hendrycks et al., 2020, MMLU), which evaluates models across 57 subjects through multiple-choice questions in fields like mathematics, history, law, and science. In our experiments, MMLU accuracy measures how well models maintain their capabilities while implementing safety defenses.
+
+Figure 2 presents our main results across models, comparing SaP against the baselines. Each defense algorithm is evaluated on all 9 attack methods over 5 seeds first, then
+
+<table><tr><td></td><td>Llama2-7B</td><td>Ministral-8B</td><td>Qwen2-1.5B</td></tr><tr><td>Polytope</td><td>91.24 ± 2.74</td><td>90.82 ± 0.23</td><td>83.96 ± 0.31</td></tr><tr><td>MLP</td><td>97.28 ± 0.19</td><td>92.90 ± 0.13</td><td>87.56 ± 0.15</td></tr><tr><td>MLP+5</td><td>91.08 ± 7.25</td><td>90.72 ± 2.31</td><td>86.12 ± 2.86</td></tr></table>
+
+Table 1: Test accuracy (\%) of different methods over 5 seeds, early-stopped to prevent overfitting. The trained models are deployed for experiments in Figure 2. Models with similar classification accuracies do not imply the same defense capability. Having a better geometric model for the representation space is essential for successful defenses.
+
+positioned on the figure by their average attack success rates (ASR) across 9 attack algorithms. As shown, SaP achieves strong defense performance while maintaining model capabilities. For Llama2-7B, it reduces the attack success rate from  $12.92\%$  to  $0.26\%$  while preserving the original MMLU accuracy  $(45.8\%$  vs.  $45.7\%)$ . This pattern holds for Ministral-8B (ASR:  $55.77\%$  to  $3.25\%$ , accuracy:  $63.4\%$  vs  $63.3\%$ ) and Qwen2-1.5B (ASR:  $27.57\%$  to  $11.81\%$ , accuracy:  $52.7\%$ ). Detailed performance of each method on Harmbench and MMLU is presented in Appendix C.2.
+
+In contrast, the baseline methods show varying effectiveness across models. Rejection Sampling achieves competitive defense performance on Llama2 and maintains accuracy on both Llama2 and Qwen2, but significantly impacts Ministrial's MMLU performance (28.5% vs original 63.4%). Other baselines like In-Context Learning and Response Check maintain accuracy but provide weaker defense, with ASRs ranging from 1.7-28.4%. These results suggest that SaP effectively balances defense capabilities and model performance across different architectures.
+
+SaP's comparison with MLP and  $\mathrm{MLP + 5}$  in Table 1, illus
+
+![](images/d28d6418d156551703b6436d2aa421d732cef96156824bea2eb70af052f0cbc0.jpg)  
+Figure 3: Comparison of average ASR for each model, with and without Concept Encoder (CE). SaP with CE consistently shows robustness against attack across models.
+
+trates the importance of having a structured and principled geometric model for the representation space of LLMs. Prior to SaP, an intuitive practice is to train an MLP with BCE loss for binary classification first, then deploy the trained MLP for steering. However, obtaining a decent classification accuracy does not necessarily lead to better defense performance. In addition, Table 1 and Figure 2 show that while classification accuracy is on par across methods, even when scaling the MLP with more layers, SaP's geometric modeling enables significant improvements compared to the MLP and  $\mathrm{MLP} + 5$  baselines.
+
+# 4.2. The Role of the Concept Encoder
+
+Enhanced defense performance. Figure 3 compares SaP variants with and without the concept encoder  $\mathrm{E}_C$  (indicated with CE) with average ASR across attack methods on HarmBench, averaged over 5 seeds. The results demonstrate significant improvements in defense performance when using the concept encoder. For Llama2, adding CE enables almost perfect defense (averaged  $0.1\%$  ASR) against all attacks, while the variant without CE shows vulnerabilities to various attacks. The improvement is particularly pronounced for Ministral, where CE reduces ASR from an average of  $51.5\%$  to  $1.7\%$ .
+
+Qwen2 shows strong baseline robustness even without CE, and adding CE maintains this performance without significant improvements. This behavior might be attributed to Qwen2's smaller feature dimensionality (1536-dimensional) compared to Ministral and Llama-2 (both 4096-dimensional). The lower-dimensional features could be inherently easier for the polytope constraints to capture safety patterns directly, making the additional abstraction from a concept encoder less crucial for effective defense. Detailed results are presented in Appendix C.2
+
+Improved interpretability. To analyze how well the facets capture different safety concepts, we use the Beaver-Tails dataset (Ji et al., 2023), which contains  $330\mathrm{k}$  annotated
+
+sentences in 14 safety categories. Each sentence is labeled as either safe or unsafe within its category. Note that we do not provide the polytope with category labels, nor hinting category information in the prompt, hence these semantic meanings are captured without direct supervision.
+
+We run both polytope variants (with and without CE) on the complete dataset to obtain facet violation patterns. For each facet  $k$  and input  $\mathbf{x}$ , we first compute the facet violation  $\left[\phi^{\top}\tilde{\mathbf{f}} -\tilde{\xi}\right]_{k}$ , then normalize each facet's violations to within [0, 1]. We calculate the mutual information between these violation scores and the labels of each safety category to see how well each facet corresponds to each safety concept. For clear comparison, we normalize each row in the resulting mutual information matrix by dividing by its maximum value, ensuring that for each safety category, the highest mutual information score is 1. More details can be found in Appendix D.1.
+
+Figure 4 compares the mutual information heatmaps with and without the concept encoder. Without the encoder, we observe significant polysemanticity in the learned facets, manifesting as high mutual information values across multiple categories for individual facets. For example, 'Facet 36' exhibits strong correlations with categories '2' to '9', all over 0.93. After introducing the concept encoder, we observe a marked reduction in polysemanticity. The mutual information matrix shows more focused correlations, with strong correlations  $(\mathrm{MI} > 0.9)$  more likely to appear in isolation rather than in clusters.
+
+To understand the semantic concepts encoded by each facet, we analyze differences in the Kullback-Leibler (KL) divergence when masking context around negative terms in BeaverTails' child abuse dataset. Results in Figure 5 reveal specialized detection patterns: 'Facet 7' demonstrates strong sensitivity to kidnapping scenarios  $(\mathrm{KLD} = 0.366)$ , 'Facet 22' primarily focuses on sexual content  $(\mathrm{KLD} = 0.733)$  and abuse-related terms  $(\mathrm{KLD} = 0.114)$ , while 'Facet 26' shows distinct activation for bullying  $(\mathrm{KLD} = 1.404)$  with secondary response to violent terms. This differentiation suggests the facets learn to capture specific categories of harmful content rather than just detecting negative terms broadly. We refer to Appendix D for additional experiments and details on this experiment.
+
+# 4.3. Ablating the Number of Constraints
+
+To understand how the number of facets affects polytope performance, we conduct two sets of experiments. For defense evaluation, we vary the facet count from 1 to 50 and measure performance against 7 HarmBench attack methods (AutoPrompt, DirectRequest, GBDA, GCG, HumanJailbreaks, PEZ, UAT), computing the mean and standard deviation of ASR across these attacks, repeated over 5 seeds. Additionally, we evaluate SaP's classification performance as the
+
+![](images/109b2b87ffb6a87274a9c4290f6a565bfa96d8a0c4148422cd19f270ef373568.jpg)  
+(a) No Concept Encoder
+
+![](images/ff2e99e3ebeabe6c80d88837a5d4df0ecaf242d25f688b8cc4e68d52412a7be0.jpg)  
+(b) With Concept Encoder
+
+![](images/e94fcd9bf36b28f4a1f71c471a79c37fe7e6b4eaf142d34b3996e9070539542c.jpg)  
+Figure 4: Mutual Information Heatmap showing the comparison between models without (a) and with Concept Encoder (b). Polytope facets learned with a concept encoder show more disentangled activations in safety classes.
+
+![](images/b1b3c637dabb519403996959002a035fa082321c39d6d8125f09327f63e8e788.jpg)  
+Figure 5: KL divergence analysis revealing semantic specialization of SaP's facets. Higher values indicate stronger activation when masking specific context words, demonstrating how each facet learns to detect distinct categories of harmful content rather than broadly negative terms.
+
+![](images/49f65728268970dff3a34c8b20233974e2553f65f6fc92c82afd506285a65b78.jpg)
+
+number of facets increase. To this end, we vary facets from 1 to 60 and evaluate on the 14 BeaverTails categories separately. We measure classification accuracy on BeaverTails' test set, reporting the mean and standard deviation of per-category accuracy across categories, repeated over 5 seeds. We emphasize that we use BeaverTails' categories only for evaluation, that is, these categories are not given to SaP during training.
+
+Figure 6 shows how the number of polytope facets affects both defense and classification performance. In terms of defense capabilities (left), increasing facet count leads to rapid improvement in safety, particularly evident in reducing attack success rates. Llama-2 achieves near-perfect defense  $(\mathrm{ASR} < 0.1\%)$  with just 20 facets, while Ministral and Qwen2 show continued improvements up to 30 facets before stabilizing. The three models exhibit different convergence patterns: Llama-2 maintains consistently low ASR after 20 facets, while Ministral and Qwen2 show slight
+
+fluctuations even with additional facets, suggesting their defense mechanisms might be more sensitive to the specific geometric arrangements of polytope constraints.
+
+For classification accuracy (right), we observe steady improvements up to approximately 30 facets, after which performance diminishes. Ministral demonstrates the most substantial initial gains, with accuracy increasing from  $76.8\%$  to  $82.5\%$  when moving from 1 to 40 facets. Llama and Qwen2 show more gradual improvements but follow a similar pattern that the performance does not significantly improve beyond certain facet counts. A comprehensive presentation of all metric values is available in Appendix D.3. We additionally conducted safety classification with a standard fully connected neural network trained using cross entropy loss, and show that SaP's classification performance does not degrade substantially compared to a fully connected neural network with an equivalent parameter count. We present these results in Appendix D.3.
+
+![](images/a22475d22ac4d77d5c44364dc8380bc0626ce9c6cd7c261f1784e54057252e6c.jpg)  
+Figure 6: Impact of polytope constraint count on model performance. Left: Average attack success rate (ASR) across 7 HarmBench attack methods, where lower values indicate better defense. Right: Average classification accuracy across 14 BeaverTails safety categories. With more facets, SaP's performance first improves, then hits a diminishing return.
+
+![](images/a54b473737dec0c499c5c14fb3bf08f885048483e445c26459c9ee0f2a22b33a.jpg)
+
+We empirically observe that many polytope facets remain inactive during training—they rarely activate on violations in the training data. This suggests potential inefficiency in the polytope loss design of CPM, where the training objective might not effectively utilize the full capacity of available parameters in the facets.
+
+# 5. Related Work
+
+Constrained and multi-objective alignment. Recent work has explored multi-objective approaches to handle competing goals in language model alignment. Similar to our steering algorithm, MCA (Fu et al., 2024) use posttraining decoding to align LLMs to different objectives. Their treatment of safety relies on "contrastive prompts", used to obtain a Pareto front for multiple objective. In contrast, SaP treats safety as constraints that must be enforced. Training-time methods like Safe-RLHF (Dai et al., 2023), SACPO (Wachi et al., 2024), RePO (Peng et al., 2024), and Constrained DPO (Liu et al., 2024) frame safety as constraints within reward learning. Other approaches, like Panacea (Zhong et al., 2024) and MODPO (Zhou et al., 2024), focus too on training-time techniques, however, like (Fu et al., 2024), they frame safety as a multiple objectives problem, instead of constraint satisfaction. The above methods require model retraining or finetuning, lack interpretable insights into learned safety concepts, and provide no mechanisms for post-deployment safety corrections. SaP addresses these challenges by learning safety constraints directly in the representation space, enabling both post-hoc control without retraining and interpretable analysis of how different safety concepts are captured through specialized polytope boundaries. This provides a complementary approach that can work alongside training-time methods.
+
+Adversarial defenses. Prior work explores four main categories of LLM safety defenses. Input filtering methods (Cao et al., 2023; Jain et al., 2023) detect and filter suspicious
+
+prompts. Input modification approaches (Yi et al., 2024; Wei et al., 2024; Xie et al., 2023; Robey et al., 2024) modify input prompts to weaken potential attacks. Both methods are vulnerable to white-box attacks, and focus only on input-level patterns rather than addressing model-internal safety representations. Output filtering (Phute et al., 2024; Zhang et al., 2024; Wang et al., 2024) uses additional language models to check for safety violations with significant computational overhead. Despite the computation overhead, large-scale training of input-output safety classifiers can yield highly promising defense results (Sharma et al., 2025). Representation-based approaches (Casper et al., 2024; Sheshadri et al., 2024; Zou et al., 2024) modify a model's internal representations for improved robustness but rarely provide interpretable insights into their safety mechanisms. We complement the limitations by introducing constraints in the model's representation space. This approach allows us to modify models' representations, opposed to modifying/filtering user inputs or model outputs. Furthermore, it facilitates treating safety concepts as constraints that should not be violated, enabling easy detection and analysis by humans, in contrast to previous representation-based approaches.
+
+AI safety and alignment. Recent work has surfaced many challenges in AI safety and alignment. Anwar et al. (2024) elucidate fundamental challenges spanning scientific understanding, development methodologies, and sociotechnical considerations in LLM safety. While Reuel et al. (2024) advocates for sophisticated technical governance frameworks, Dalrymple et al. (2024) makes a case for rigorous approaches to design AI systems with quantifiable safety assurances. AI Control (Greenblatt et al., 2023) proposes that safety can be maintained through systems that prevent harmful actions, even when models are misaligned. In this context, SaP's geometric approach to constraints in the representation space offers a promising framework for advancing both alignment and control.
+
+# 6. Limitations and Future Work
+
+Our results encourage further practical and theoretical development of SaP. A key future work direction lies in further disentangling semantic understandings across polytope edges. CPM represents progress, however, its heuristic facet assignment algorithm limits its overall performance. Leveraging tools from mechanistic interpretability (Black et al., 2022) or exploring other (possibly non-linear) geometric representations of safety (Park et al., 2024) could reveal deeper insights into the learned constraints.
+
+Based on our novel geometric framework for explicit safety modeling, SaP demonstrates an effective defense against various adversarial attacks. However, we observe that under many attacks, specifically for the Ministrial-8B model, SaP can induce semantically incoherent outputs. Importantly, all evaluated models, including Ministrial-8B, perform as expected on standard capability benchmarks like MMLU and MT-Bench (Bai et al., 2024), with results listed in Tables 8 and 9. Although comprehensive adversarial robustness and broad capability improvements are beyond the scope of this paper, we encourage future research to investigate and improve SaP's performance in these critical domains.
+
+In terms of theory, our work lacks a more rigorous understanding of SaP. Current sample complexity bounds for learning polytopes build on Probably Approximately Correct (PAC) learning. The PAC-style results of Gottlieb et al. (2018) provide a solid starting point, as they shed light on the fat-shattering dimension required to learn polytopes (discussed in Appendix E). However, these results depend on strong assumptions that are hard to verify in practice. Adapting these results to LLMs could provide further insights into what can be achieved in terms of formal guarantees for LLM safety, as stressed by Dalrymple et al. (2024).
+
+# 7. Conclusion
+
+We introduce SaP, a geometric framework for learning safety constraints in LLMs. Motivated by constrained Markov decision processes, we formulate safety as a geometric problem, introducing constraints in model representations. This view enables post-deployment control without model fine-tuning and provides interpretable insights into the learned safety mechanism. Experiment results demonstrate the effectiveness of this framework in defending against adversarial attacks while maintaining model capabilities and offering facets that naturally correspond to different safety concepts. While significant challenges remain in developing safe AI, this geometric perspective offers a promising framework for aligning and controlling model behaviors. Substantial research remains necessary, from developing more sophisticated geometric frameworks to establishing rigorous theoretical foundations for safety guarantees in large-scale models.
+
+# Acknowledgement
+
+We thank Javier Rando and Daniel Paleka for their valuable input during the development of this framework. We are grateful to the anonymous reviewers for their constructive feedback, which led to additional experiments and baselines that strengthened our arguments. We also thank Ido Hakimi, Vignesh Ram Somnath and Roni Globerman for their helpful comments on earlier versions of the manuscript. Xin Chen is supported by the Open Philanthropy AI Fellowship and the Vitalik Buterin Fellowship from the Future of Life Institute. Yarden As is supported by the ETH AI Center Fellowship and a grant of the Hasler foundation (no. 21039). The research received further support through ELSA (European Lighthouse on Secure and Safe AI) funded by the European Union under grant agreement No. 101070617.
+
+# Impact Statement
+
+The increasing capabilities and deployment of large language models (LLMs) highlight the critical importance of ensuring their safety. LLMs can generate harmful content and are vulnerable to adversarial attacks, posing significant risks. Research focused on developing robust safety measures is essential to mitigate these risks and promote the responsible development of LLMs. This work contributes to advancing the field of LLM safety by introducing a novel geometric framework for learning and enforcing safety constraints. We believe that our research can support the creation of safer and more reliable AI systems, ultimately benefiting society by reducing the potential for harm.
+
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+
+# A. Entropy-based Heuristic Facet Assignment Strategy
+
+Following the Convex Polytope Machine (Kantchelian et al., 2014) algorithm, we use an entropy-based heuristic strategy to select constraint violations while preventing overreliance on individual facets. For each unsafe example  $x$  where  $y = -1$ , rather than simply choosing the facet  $k$  with maximum violation  $\max_k (\phi_k^\top \tilde{\mathbf{f}} - \tilde{\xi}_k)$ , the method considers the set of valid facets  $\mathcal{K}_{\mathrm{valid}}$  whose violations exceed a threshold  $\tau$ :
+
+$$
+\mathcal {K} _ {\text {v a l i d}} = \left\{k: \phi_ {k} ^ {\top} \tilde {\mathbf {f}} - \tilde {\xi} _ {k} > \tau \right\} \tag {3}
+$$
+
+The entropy of facet assignments is computed across the set of unsafe examples  $\mathcal{U}$  as:
+
+$$
+H (\mathcal {K}) = - \sum_ {k = 1} ^ {K} p _ {k} \log_ {2} p _ {k} \tag {4}
+$$
+
+where  $p_k = \frac{|\{x \in \mathcal{U} : k = z(x)\}|}{|\mathcal{U}|}$  is the proportion of unsafe examples assigned to facet  $k$ .
+
+Algorithm 2 Entropy-based facet Assignment  
+Require: Feature matrix  $\tilde{\mathbf{f}}$  labels  $y$  constraint matrix  $\phi$  ,thresholds  $\tilde{\xi}$    
+Require: Valid facet threshold  $\tau$  entropy threshold  $h_\mathrm{target}$  max attempts  $T$    
+1:  $\mathcal{U}\gets \{i:y_i = -1\}$    
+2:  $k_{i}\gets \arg \max_{k}(\phi_{k}^{\top}\tilde{\mathbf{f}}_{i} - \tilde{\xi}_{k})$  for all  $i\in \mathcal{U}$    
+3:  $H\gets$  ComputeEntropy(  $\{k_i\}_{i\in \mathcal{U}}$    
+4: while  $H <   h_{\mathrm{target}}$  and attempts  $<  T$  do   
+5: Sample  $i$  randomly from U   
+6:  $\mathcal{K}_{\mathrm{valid}}\gets \{k:\phi_k^\top \tilde{\mathbf{f}}_i - \tilde{\xi}_k > \tau \}$    
+7: Sample  $k_{\mathrm{new}}$  randomly from  $\mathcal{K}_{\mathrm{valid}}$    
+8:  $H_{\mathrm{new}}\gets$  ComputeEntropy(  $\{k_i\}_{i\in \mathcal{U}}\cup \{k_{\mathrm{new}}\}$    
+9: if  $H_{\mathrm{new}} > H$  then   
+10:  $k_{i}\gets k_{\mathrm{new}}$    
+11:  $H\gets H_{\mathrm{new}}$    
+12: end if   
+13: end while   
+output Facet assignments  $\{k_i\}_{i\in \mathcal{U}}$
+
+Set of unsafe examples Initial assignments
+
+The algorithm iteratively attempts to increase assignment entropy by randomly selecting alternative facets from  $\mathcal{K}_{\mathrm{valid}}$  while maintaining strong constraint violations. The target entropy threshold  $h_{\mathrm{target}}$  is heuristically set to  $\frac{1}{2}\log_2K$ , where  $K$  is the total number of facets.  $\tau$  is set to 0. This approach ensures diverse facet utilization across unsafe examples while preserving the effectiveness of the safety constraints.
+
+# B. Steering via Lagrangian Relaxation
+
+Building on the learned polytope facets, SaP implements directional steering in the model's representation space to constrain generation while minimizing disruption to the model's capabilities. This approach enables fine-grained safety control through local activation editing, avoiding the need for model fine-tuning or prompt engineering.
+
+One way to solve the constrained problem in Equation (2) is through a Lagrangian relaxation. To this end, in practice, we perform 100 gradient steps on the following loss function
+
+$$
+\min  _ {\boldsymbol {h}} \left\| \bar {\pi} _ {l} (\boldsymbol {x}) - \boldsymbol {h} \right\| _ {1} + \lambda_ {\text {s a f e}} \sum_ {k = 1} ^ {K} \left[ \phi_ {k} ^ {\top} \mathrm {E} _ {C} (\boldsymbol {h}) - \tilde {\xi} _ {k} \right] _ {-} + \lambda_ {\text {u n s a f e}} \sum_ {k = 1} ^ {K} \left[ \phi_ {k} ^ {\top} \mathrm {E} _ {C} (\boldsymbol {h}) - \tilde {\xi} _ {k} \right] _ {+}
+$$
+
+where  $h$  is the edited activation,  $\mathrm{E}_C(h)$  is its corresponding encoded feature,  $[\cdot]_+ = \max(0, \cdot)$  denotes the positive part, and  $[\cdot]_- = \min(0, \cdot)$  denotes the negative part. The hyperparameters  $\lambda_{\text{safe}}$  and  $\lambda_{\text{unsafe}}$  control the penalties for negative and positive violations respectively, allowing for asymmetric treatment of constraint violations.
+
+# C. Harmbench Training Setup and Results
+
+# C.1. Harmbench Training Setup
+
+In Harmbench (Mazeika et al., 2024), we save the per-token representation at layer 20 during the original model output generation (with attacked inputs). The total training dataset size is num_sentences  $\times$  tokens_per_sentence. If an output sentence is marked as unsafe, we label all token representations in the sentence as unsafe, and vice versa. For each model (Llama, Mistral, Qwen), we use its default system prompt template when inferencing both the original and the steered model.
+
+For the polytope training phase, we use the Adam optimizer with a learning rate of  $10^{-2}$  and batch size of 128. The feature extractor projects hidden states to a 16,384-dimensional space followed by ReLU activation. The loss function uses an entropy weight 1.0 and feature L1 regularization weight  $\lambda_{\phi} = 1.0$ . The margin parameter  $\kappa$  varies across model architectures: 60.0 for Llama-2 7B, 5.0 for Ministrial 8B, and 30.0 for Qwen2 1.5B, reflecting different geometric requirements in their respective representation spaces.
+
+During the steering phase, we apply hidden state optimization at layer 20 with model-specific configurations. For Llama-2 7B, we set  $\lambda_{\mathrm{unsafe}} = 4.0$  and  $\lambda_{\mathrm{safe}} = 10^{-4}$  for the optimization objective. Ministral 8B uses  $\lambda_{\mathrm{unsafe}} = 0.25$  without safety violation penalty ( $\lambda_{\mathrm{safe}} = 0$ ). For Qwen2 1.5B, we set  $\lambda_{\mathrm{unsafe}} = 10.0$  and  $\lambda_{\mathrm{safe}} = 5000$ . All models are quantized to 16-bit precision (float16 for Llama-2 and bffloat16 for Ministral and Qwen) to reduce memory requirements while maintaining numerical stability.
+
+# C.2. Harmbench Main Results
+
+Figure 2 is created by aggregating the average of each defense method's performance over all 7 attack algorithms. We present detailed Llama-2 7B results in Table 2 and Table 3, Ministral-8B results in Table 4 and Table 5, and Qwen2-1.5B results in Table 6 and Table 7. We perform experiments in the same setup with prompt-based baselines Self Reminder (Xie et al., 2023), Response Check (Wang et al., 2024), and SmoothLLM (Robey et al., 2024), implemented in the BackTranslation code base (Wang et al., 2024).
+
+We also implemented rejection sampling, which uses the polytope classifier. Instead of steering when a token is detected unsafe, it immediately modifies the current token output to be "Sorry". It empirically works well for defense, but might result in over-rejecting reasonable requests, as shown in the MMLU results (Table 8). All methods' results (mean ± standard deviation) are obtained over 5 seeds. Table 10 shows results over 7 attack algorithms for each model, when trained with and without the concept encoder.
+
+Table 2: Defense comparison for Llama2-7B across different attack methods. Lower values indicate better defense effectiveness (0 means perfect defense). Each method is evaluated over 5 seeds.  
+
+<table><tr><td>Attack Method</td><td>Original</td><td>Steering</td><td>Rejection</td><td>Response Check</td><td>Self Reminder</td><td>SmoothLLM</td></tr><tr><td>AutoDAN</td><td>1.77</td><td>0.00 ± 0.00</td><td>0.00 ± 0.00</td><td>0.51 ± 0.00</td><td>0.00 ± 0.00</td><td>2.33 ± 0.33</td></tr><tr><td>AutoPrompt</td><td>16.5</td><td>0.00 ± 0.00</td><td>0.05 ± 0.11</td><td>8.25 ± 0.00</td><td>1.25 ± 0.00</td><td>10.30 ± 1.58</td></tr><tr><td>DirectRequest</td><td>1.0</td><td>0.00 ± 0.00</td><td>0.05 ± 0.11</td><td>0.50 ± 0.00</td><td>0.00 ± 0.00</td><td>6.50 ± 0.64</td></tr><tr><td>GBDA</td><td>0</td><td>0.00 ± 0.00</td><td>0.00 ± 0.00</td><td>0.06 ± 0.04</td><td>0.00 ± 0.00</td><td>4.21 ± 0.39</td></tr><tr><td>GCG</td><td>30</td><td>0.40 ± 0.29</td><td>0.10 ± 0.22</td><td>13.00 ± 0.00</td><td>2.50 ± 0.00</td><td>12.10 ± 0.68</td></tr><tr><td>HumanJailbreaks</td><td>0.5</td><td>0.00 ± 0.00</td><td>0.01 ± 0.02</td><td>0.09 ± 0.01</td><td>0.05 ± 0.00</td><td>2.19 ± 0.45</td></tr><tr><td>PEZ</td><td>0</td><td>0.00 ± 0.00</td><td>0.00 ± 0.00</td><td>0.00 ± 0.00</td><td>0.00 ± 0.00</td><td>4.03 ± 0.48</td></tr><tr><td>UAT</td><td>6.5</td><td>0.00 ± 0.00</td><td>0.00 ± 0.00</td><td>2.75 ± 0.00</td><td>0.50 ± 0.00</td><td>8.45 ± 0.65</td></tr><tr><td>AdaptiveAttack</td><td>60</td><td>2.00 ± 0.00</td><td>0.00 ± 0.00</td><td>0.00 ± 0.00</td><td>2.00 ± 0.00</td><td>2.04 ± 0.00</td></tr></table>
+
+Table 3: MLP defense comparison for Llama2-7B across different attack methods, evaluated over 5 seeds. Lower values indicate better defense effectiveness.  
+
+<table><tr><td>Attack Method</td><td>Original</td><td>MLP</td><td>MLP+5</td></tr><tr><td>AutoDAN</td><td>1.77</td><td>0.25 ± 0.00</td><td>0.25 ± 0.00</td></tr><tr><td>AutoPrompt</td><td>16.5</td><td>12.80 ± 0.27</td><td>12.85 ± 0.14</td></tr><tr><td>DirectRequest</td><td>1.0</td><td>0.30 ± 0.11</td><td>0.25 ± 0.00</td></tr><tr><td>GBDA</td><td>0</td><td>0.00 ± 0.00</td><td>0.00 ± 0.00</td></tr><tr><td>GCG</td><td>30</td><td>27.10 ± 0.29</td><td>26.90 ± 0.29</td></tr><tr><td>HumanJailbreaks</td><td>0.5</td><td>0.34 ± 0.26</td><td>0.24 ± 0.02</td></tr><tr><td>PEZ</td><td>0</td><td>0.00 ± 0.00</td><td>0.00 ± 0.00</td></tr><tr><td>UAT</td><td>6.5</td><td>4.75 ± 0.18</td><td>4.75 ± 0.00</td></tr><tr><td>AdaptiveAttack</td><td>60</td><td>0.00 ± 0.00</td><td>0.00 ± 0.00</td></tr></table>
+
+Table 4: Defense comparison for Mistral-8B across different attack methods. Lower values indicate better defense effectiveness (0 means perfect defense). Each method is evaluated over 5 seeds.  
+
+<table><tr><td>Attack Method</td><td>Original</td><td>Steering</td><td>Rejection</td><td>Response Check</td><td>Self Reminder</td><td>SmoothLLM</td></tr><tr><td>AutoDAN</td><td>66.75</td><td>1.35 ± 0.49</td><td>0.00 ± 0.00</td><td>1.50 ± 0.00</td><td>11.50 ± 0.00</td><td>19.38 ± 1.94</td></tr><tr><td>AutoPrompt</td><td>48.3</td><td>2.79 ± 1.64</td><td>0.51 ± 0.25</td><td>7.88 ± 0.00</td><td>13.94 ± 0.00</td><td>40.36 ± 2.63</td></tr><tr><td>DirectRequest</td><td>42.8</td><td>1.20 ± 0.54</td><td>0.33 ± 0.13</td><td>5.25 ± 0.00</td><td>2.50 ± 0.00</td><td>16.30 ± 2.18</td></tr><tr><td>GBDA</td><td>62.8</td><td>1.26 ± 1.10</td><td>0.10 ± 0.23</td><td>14.62 ± 4.37</td><td>6.75 ± 0.00</td><td>36.75 ± 8.71</td></tr><tr><td>GCG</td><td>67.5</td><td>1.90 ± 1.10</td><td>0.08 ± 0.13</td><td>5.75 ± 0.00</td><td>26.50 ± 0.00</td><td>32.35 ± 2.00</td></tr><tr><td>HumanJailbreaks</td><td>39.4</td><td>1.15 ± 0.68</td><td>0.01 ± 0.02</td><td>2.36 ± 0.13</td><td>9.65 ± 0.01</td><td>17.41 ± 0.88</td></tr><tr><td>PEZ</td><td>33.6</td><td>1.14 ± 0.40</td><td>0.00 ± 0.00</td><td>11.38 ± 4.79</td><td>1.97 ± 0.00</td><td>27.00 ± 10.63</td></tr><tr><td>UAT</td><td>40.8</td><td>2.45 ± 1.04</td><td>0.54 ± 0.25</td><td>6.25 ± 0.00</td><td>4.50 ± 0.00</td><td>24.65 ± 3.09</td></tr><tr><td>AdaptiveAttack</td><td>100</td><td>14.00 ± 0.00</td><td>0.00 ± 0.00</td><td>97.33 ± 0.94</td><td>90.00 ± 0.00</td><td>97.33 ± 0.94</td></tr></table>
+
+Table 5: MLP defense comparison for Ministrial-8B across different attack methods, evaluated over 5 seeds. Lower values indicate better defense effectiveness.  
+
+<table><tr><td>Attack Method</td><td>Original</td><td>MLP</td><td>MLP+5</td></tr><tr><td>AutoDAN</td><td>66.75</td><td>16.90 ± 5.14</td><td>17.80 ± 1.34</td></tr><tr><td>AutoPrompt</td><td>48.3</td><td>74.30 ± 2.04</td><td>73.58 ± 1.58</td></tr><tr><td>DirectRequest</td><td>42.8</td><td>20.50 ± 1.06</td><td>20.35 ± 1.29</td></tr><tr><td>GBDA</td><td>62.8</td><td>66.31 ± 22.82</td><td>80.91 ± 3.12</td></tr><tr><td>GCG</td><td>67.5</td><td>60.35 ± 0.63</td><td>61.10 ± 0.82</td></tr><tr><td>HumanJailbreaks</td><td>39.4</td><td>27.96 ± 8.03</td><td>25.62 ± 1.75</td></tr><tr><td>PEZ</td><td>33.6</td><td>32.72 ± 12.77</td><td>32.81 ± 15.03</td></tr><tr><td>UAT</td><td>40.8</td><td>49.90 ± 1.07</td><td>49.25 ± 1.38</td></tr><tr><td>AdaptiveAttack</td><td>100</td><td>16.00 ± 0.00</td><td>16.00 ± 0.00</td></tr></table>
+
+# D. BeaverTails Training Setup and Results
+
+# D.1. Mutual Information Calculation Details
+
+For BeaverTails (Ji et al., 2023), the Mutual Information (MI) between each safety category and each facet's violations are computed using scikit-learn's mutual_information_score function. We compare the facets training from two polytopes, both share the same set of hyperparameters, and the only difference is whether the Concept Encoder is included in the training. They are both trained with 50 facets, trained on the full BeaverTails dataset over 20 epochs, optimized with Adam optimizer with learning rate 0.01, batch size 128, and  $\lambda_{\phi} 0.01$ . The polytope trained with Concept Encoder has an encoded feature dimension 16384,  $\lambda_{\tilde{F}} 1.0$ , and a margin 2.0.
+
+Table 6: Defense comparison for Qwen-1.5B across different attack methods. Lower values indicate better defense effectiveness (0 means perfect defense). Each method is evaluated over 5 seeds.  
+
+<table><tr><td>Attack Method</td><td>Original</td><td>Steering</td><td>Rejection</td><td>Response Check</td><td>Self Reminder</td><td>SmoothLLM</td></tr><tr><td>AutoDAN</td><td>45.00</td><td>0.80 ± 0.82</td><td>0.67 ± 0.00</td><td>38.25 ± 0.00</td><td>30.50 ± 0.00</td><td>6.85 ± 1.27</td></tr><tr><td>AutoPrompt</td><td>17.90</td><td>1.16 ± 0.55</td><td>2.83 ± 1.09</td><td>23.74 ± 0.00</td><td>3.54 ± 0.00</td><td>6.77 ± 1.20</td></tr><tr><td>DirectRequest</td><td>13.25</td><td>0.15 ± 0.22</td><td>0.60 ± 0.29</td><td>5.00 ± 0.00</td><td>2.00 ± 0.00</td><td>5.60 ± 1.04</td></tr><tr><td>GBDA</td><td>19.45</td><td>1.42 ± 0.87</td><td>3.02 ± 0.52</td><td>20.63 ± 2.41</td><td>3.40 ± 0.00</td><td>8.54 ± 0.63</td></tr><tr><td>GCG</td><td>25.00</td><td>1.00 ± 0.69</td><td>2.64 ± 2.43</td><td>45.00 ± 0.00</td><td>22.86 ± 0.00</td><td>11.43 ± 1.16</td></tr><tr><td>HumanJailbreaks</td><td>4.14</td><td>0.18 ± 0.13</td><td>0.28 ± 0.23</td><td>5.40 ± 0.00</td><td>4.00 ± 0.00</td><td>4.42 ± 0.50</td></tr><tr><td>PEZ</td><td>10.90</td><td>0.45 ± 0.44</td><td>1.06 ± 0.74</td><td>4.85 ± 0.00</td><td>0.75 ± 0.00</td><td>4.63 ± 0.48</td></tr><tr><td>UAT</td><td>12.50</td><td>1.15 ± 0.52</td><td>2.70 ± 1.14</td><td>14.75 ± 0.00</td><td>1.75 ± 0.00</td><td>7.30 ± 1.56</td></tr><tr><td>AdaptiveAttack</td><td>100</td><td>100.00 ± 0.00</td><td>90.00 ± 0.00</td><td>100.00 ± 0.00</td><td>100.00 ± 0.00</td><td>100.00 ± 0.00</td></tr></table>
+
+Table 7: MLP defense comparison for Qwen-1.5B across different attack methods, evaluated over 5 seeds. Lower values indicate better defense effectiveness.  
+
+<table><tr><td>Attack Method</td><td>Original</td><td>MLP</td><td>MLP+5</td></tr><tr><td>AutoDAN</td><td>45.00</td><td>3.75 ± 1.56</td><td>12.85 ± 14.36</td></tr><tr><td>AutoPrompt</td><td>17.90</td><td>8.23 ± 1.62</td><td>14.29 ± 12.29</td></tr><tr><td>DirectRequest</td><td>13.25</td><td>3.10 ± 0.63</td><td>4.00 ± 2.67</td></tr><tr><td>GBDA</td><td>19.45</td><td>9.92 ± 1.44</td><td>16.80 ± 11.80</td></tr><tr><td>GCG</td><td>25.00</td><td>10.57 ± 3.35</td><td>18.43 ± 21.13</td></tr><tr><td>HumanJailbreaks</td><td>4.14</td><td>2.26 ± 0.55</td><td>4.17 ± 4.38</td></tr><tr><td>PEZ</td><td>10.9</td><td>2.81 ± 0.50</td><td>5.07 ± 2.21</td></tr><tr><td>UAT</td><td>12.5</td><td>6.75 ± 1.08</td><td>10.95 ± 6.36</td></tr><tr><td>AdaptiveAttack</td><td>100</td><td>100.00 ± 0.00</td><td>100.00 ± 0.00</td></tr></table>
+
+Table 8: MMLU accuracy (%) comparison across different defense methods. Results show that most defense methods maintain original model performance, with some exceptions for rejection and self-reminder strategies.  
+
+<table><tr><td>Method</td><td>Llama2-7B</td><td>Ministral-8B</td><td>Qwen2-1.5B</td></tr><tr><td>Original</td><td>45.8</td><td>63.4</td><td>52.7</td></tr><tr><td>Steering</td><td>45.7</td><td>63.3</td><td>52.7</td></tr><tr><td>Rejection</td><td>44.6</td><td>28.5</td><td>51.2</td></tr><tr><td>ICL</td><td>45.8</td><td>65.1</td><td>55.4</td></tr><tr><td>Response Check</td><td>45.8</td><td>65.1</td><td>55.4</td></tr><tr><td>Self Reminder</td><td>45.8</td><td>22.9</td><td>22.9</td></tr><tr><td>SmoothLLM</td><td>45.8</td><td>65.1</td><td>55.4</td></tr><tr><td>MLP</td><td>45.8</td><td>63.4</td><td>55.4</td></tr><tr><td>MLP+5</td><td>45.8</td><td>63.4</td><td>55.4</td></tr></table>
+
+# D.2. BeaverTails Training Hyperparameters
+
+We conduct polytope training on all BeaverTails classes. We first obtain the (human question, model answer) pairs from the dataset, formatting the input as f" {humanQuestion} \n{model_answer}" using Python f-string without any other prompts. We observe that this works better for classifying safety concepts than other basic types of input prompting. We then obtain the activation for each sentence at a model's layer  $l$  at the sentence's last word.
+
+The polytope model is optimized using Adam with a learning rate of  $10^{-2}$  and batch size of 128. For facet assignment, we use a valid facet threshold  $\tau = 0.0$ . The concept encoder uses a feature dimension of 16,384. The training runs for 20 epochs.
+
+Table 9: Model performance across turns in MT-Bench. There are fluctuations in LLM-based judges' scores, but all models produce coherent and high-quality sentences, without signs of rejections.  
+
+<table><tr><td>Model</td><td>First Turn</td><td>Second Turn</td><td>Average</td></tr><tr><td>Llama2-7B</td><td>6.54</td><td>5.46</td><td>6.00</td></tr><tr><td>Llama2-7B + SaP</td><td>6.98</td><td>6.21</td><td>6.59</td></tr><tr><td>Ministral-8B</td><td>7.55</td><td>6.98</td><td>7.26</td></tr><tr><td>Ministral-8B + SaP</td><td>9.01</td><td>7.64</td><td>8.33</td></tr><tr><td>Qwen-1.5B</td><td>6.90</td><td>5.15</td><td>6.03</td></tr><tr><td>Qwen-1.5B + SaP</td><td>6.29</td><td>4.70</td><td>5.49</td></tr></table>
+
+Table 10: Attack success rate comparison between models, with and without Concept Encoder. Results show mean ± standard deviation across different attack methods.  
+
+<table><tr><td rowspan="2">Method</td><td colspan="2">Llama-2 7B</td><td colspan="2">Ministral-8B</td><td colspan="2">Qwen2 1.5B</td></tr><tr><td>Without CE</td><td>With CE</td><td>Without CE</td><td>With CE</td><td>Without CE</td><td>With CE</td></tr><tr><td>AutoPrompt</td><td>10.7 ± 0.0</td><td>0.00 ± 0.00</td><td>73.3 ± 0.0</td><td>2.79 ± 1.64</td><td>0.45 ± 0.33</td><td>1.16 ± 0.55</td></tr><tr><td>DirectRequest</td><td>0.0 ± 0.0</td><td>0.00 ± 0.00</td><td>21.5 ± 0.0</td><td>1.20 ± 0.54</td><td>0.15 ± 0.14</td><td>0.15 ± 0.22</td></tr><tr><td>GBDA</td><td>0.0 ± 0.0</td><td>0.00 ± 0.00</td><td>82.3 ± 0.0</td><td>1.26 ± 1.10</td><td>0.71 ± 0.28</td><td>1.42 ± 0.87</td></tr><tr><td>GCG</td><td>26.5 ± 0.0</td><td>0.40 ± 0.29</td><td>60.8 ± 0.0</td><td>1.90 ± 1.10</td><td>0.57 ± 0.54</td><td>1.00 ± 0.69</td></tr><tr><td>HumanJailbreaks</td><td>0.3 ± 0.0</td><td>0.00 ± 0.00</td><td>24.7 ± 0.0</td><td>1.15 ± 0.68</td><td>0.08 ± 0.08</td><td>0.18 ± 0.13</td></tr><tr><td>PEZ</td><td>0.0 ± 0.0</td><td>0.00 ± 0.00</td><td>48.0 ± 0.0</td><td>1.14 ± 0.40</td><td>0.32 ± 0.07</td><td>0.45 ± 0.44</td></tr><tr><td>UAT</td><td>4.0 ± 0.0</td><td>0.00 ± 0.00</td><td>50.2 ± 0.0</td><td>2.45 ± 1.04</td><td>0.45 ± 0.33</td><td>1.15 ± 0.52</td></tr></table>
+
+![](images/137d8634995927c6ba89cb2c376d9694df92d482b4d04e5285ce5f6084ad2009.jpg)
+
+![](images/f060cf3c0b3a6b1d77446caaa31bc2ba8e45e73b9a82a7d6d524ea6d0abc1140.jpg)
+
+![](images/5e9a05c2d97d7c63e5d702be0356d4c1d75c4f7f27b555182e0ecf4eef2bc5c5.jpg)
+
+![](images/2004f94314d1ecdd14525aaa8869505d63da46a9c505b2a0050a16e9279a25b9.jpg)  
+Figure 7: KL divergence analysis revealing semantic specialization of SaP's facets on BeaverTail's terrorism dataset. Facets can activate at one single concept (top row plots), or multiple concepts (bottom row plots)
+
+![](images/2dfb5a79b042de031a02ed4fb42faa15b59448c7a8f1fef99754ec778491e6c0.jpg)
+
+![](images/e69b74d05ba55931978956d58b29f13dcb230ab0cc4f24eb17b4209fe2a082a7.jpg)
+
+We empirically find that margin parameter  $\kappa$  and the polytope sparsity weight  $\lambda_{\phi}$  significantly influence model performance. Different safety categories and model architectures require different hyperparameter values for optimal performance. Table 11 shows hyperparameters for Llama2-7B, and Table 12 and 13 show hyperparameters for Ministral-8B and Qwen2-1.5B respectively.
+
+Table 11: Category-specific hyperparameters for BeaverTails for Llama2-7B. The last row "All data" shows the hyperparameters used to train all BeaverTails data.  
+
+<table><tr><td>Category</td><td>Margin</td><td>λφ</td></tr><tr><td>Animal abuse</td><td>0.5</td><td>0.01</td></tr><tr><td>Child abuse</td><td>0.7</td><td>0.01</td></tr><tr><td>Politics</td><td>3.0</td><td>0.0001</td></tr><tr><td>Discrimination</td><td>2.0</td><td>0.0001</td></tr><tr><td>Drug abuse</td><td>1.0</td><td>0.01</td></tr><tr><td>Financial crime</td><td>2.0</td><td>0.01</td></tr><tr><td>Hate speech</td><td>10.0</td><td>0.0001</td></tr><tr><td>Misinformation</td><td>0.8</td><td>0.0001</td></tr><tr><td>Unethical behavior</td><td>20.0</td><td>0.0001</td></tr><tr><td>Privacy violation</td><td>1.0</td><td>0.01</td></tr><tr><td>Self-harm</td><td>1.0</td><td>0.0001</td></tr><tr><td>Adult content</td><td>1.0</td><td>0.01</td></tr><tr><td>Terrorism</td><td>1.0</td><td>0.01</td></tr><tr><td>Violence</td><td>20.0</td><td>0.0001</td></tr><tr><td>All data</td><td>20.0</td><td>0.01</td></tr></table>
+
+Table 12: Category-specific hyperparameters for BeaverTails for Ministrial-8B. The last row "All data" shows the hyperparameters used to train all BeaverTails data.  
+
+<table><tr><td>Category</td><td>Margin</td><td>λφ</td></tr><tr><td>Animal abuse</td><td>1.0</td><td>0.0001</td></tr><tr><td>Child abuse</td><td>0.8</td><td>0.0001</td></tr><tr><td>Politics</td><td>10.0</td><td>0.0001</td></tr><tr><td>Discrimination</td><td>10.0</td><td>0.0001</td></tr><tr><td>Drug abuse</td><td>2.0</td><td>0.0001</td></tr><tr><td>Financial crime</td><td>2.0</td><td>0.0001</td></tr><tr><td>Hate speech</td><td>1.0</td><td>0.0001</td></tr><tr><td>Misinformation</td><td>10.0</td><td>0.0001</td></tr><tr><td>Unethical behavior</td><td>15.0</td><td>0.0001</td></tr><tr><td>Privacy violation</td><td>2.0</td><td>0.0001</td></tr><tr><td>Self-harm</td><td>2.0</td><td>0.01</td></tr><tr><td>Adult content</td><td>0.6</td><td>0.01</td></tr><tr><td>Terrorism</td><td>0.6</td><td>0.01</td></tr><tr><td>Violence</td><td>2.0</td><td>0.01</td></tr><tr><td>All data</td><td>20.0</td><td>0.0001</td></tr></table>
+
+# D.3. BeaverTails Results
+
+We present test accuracy results across different configurations of facets of Llama2-7B, Ministral-8B, and Qwen2-1.5B in Tables 14, 15, and 16 respectively. For each category, we additionally run learning on a multi-layer perceptron (MLP) classifier using the same representation inputs. For fair comparison, the MLP is defined to have a similar model size with the polytope trained for each safety category. The MLP architecture has two layers and a classification head:
+
+- The first layer is  $\mathrm{ReLU}(W_1\bar{\pi}_l(\pmb{x}) + b_1)$ , with  $\bar{\pi}_l(\pmb{x})$  being the LLM activation at layer  $l$  (i.e., the same representation input as the one used in polytope training). This layer has the same number of parameters as the concept encoder, with  $W_1 \in \mathbb{R}^{16384 \times d_l}$ ,  $b_1 \in \mathbb{R}^{16384}$ .  
+- We then add a second layer  $\mathrm{ReLU}(W_2x_1 + b_2)$ , with  $W_{2}\in \mathbb{R}^{n_{\mathrm{facet}}\times 16384}$ , with  $x_{1}$  being the output of the first layer,  $b_{1}\in \mathbb{R}^{n_{\mathrm{facet}}}$  matching the parameter size of the polytope  $\phi$  and  $\tilde{\xi}$ .
+
+Table 13: Category-specific hyperparameters for BeaverTails for Qwen-1.5B. The last row "All data" shows the hyperparameters used to train all BeaverTails data.  
+
+<table><tr><td>Category</td><td>Margin</td><td>λφ</td></tr><tr><td>Animal abuse</td><td>15.0</td><td>0.0001</td></tr><tr><td>Child abuse</td><td>2.0</td><td>0.01</td></tr><tr><td>Politics</td><td>20.0</td><td>0.0001</td></tr><tr><td>Discrimination</td><td>10.0</td><td>0.0001</td></tr><tr><td>Drug abuse</td><td>15.0</td><td>0.0001</td></tr><tr><td>Financial crime</td><td>10.0</td><td>0.0001</td></tr><tr><td>Hate speech</td><td>0.8</td><td>0.01</td></tr><tr><td>Misinformation</td><td>0.8</td><td>0.01</td></tr><tr><td>Unethical behavior</td><td>15.0</td><td>0.0001</td></tr><tr><td>Privacy violation</td><td>2.0</td><td>0.0001</td></tr><tr><td>Self-harm</td><td>2.0</td><td>0.0001</td></tr><tr><td>Adult content</td><td>3.0</td><td>0.0001</td></tr><tr><td>Terrorism</td><td>0.9</td><td>0.0001</td></tr><tr><td>Violence</td><td>10.0</td><td>0.0001</td></tr><tr><td>All data</td><td>10.0</td><td>0.01</td></tr></table>
+
+- To perform classification, MLP has an additional classification head defined by  $\operatorname{Softmax}(W_3x_2 + b_3)$ , where  $x_2$  is the second layer output,  $W_3 \in \mathbb{R}^{2 \times n_{\mathrm{facet}}}$ ,  $b_3 \in \mathbb{R}^2$ . It outputs the model's estimated probability whether an input is safe or unsafe (i.e., 2 classes). Hence, for each category, the MLP has  $n_{\mathrm{facet}} \times 2 + 2$  more parameters than the SaP polytope.
+
+The MLP is trained using cross-entropy loss and an Adam optimizer with learning rate 0.0001. We present the results in Table 17. More KL divergence analysis on SaP's facets are presented in Table 7.
+
+Table 14: Test accuracy (%) comparison across different numbers of facets for each category in Llama2-7B. Results show mean ± standard deviation. Bold numbers indicate the selected configuration in our final model.  
+
+<table><tr><td>Category</td><td>1 facet</td><td>10 facets</td><td>20 facets</td><td>30 facets</td><td>40 facets</td><td>50 facets</td><td>60 facets</td></tr><tr><td>Animal abuse</td><td>60.0 ± 18.0</td><td>92.0 ± 3.0</td><td>93.0 ± 1.4</td><td>92.0 ± 3.0</td><td>91.0 ± 4.0</td><td>93.0 ± 1.0</td><td>89.0 ± 2.0</td></tr><tr><td>Child abuse</td><td>85.0 ± 8.0</td><td>91.8 ± 2.2</td><td>89.0 ± 4.0</td><td>88.0 ± 1.0</td><td>88.0 ± 2.0</td><td>87.0 ± 2.0</td><td>89.0 ± 3.0</td></tr><tr><td>Politics</td><td>82.0 ± 3.0</td><td>82.0 ± 1.0</td><td>82.0 ± 2.0</td><td>84.1 ± 0.5</td><td>80.0 ± 2.0</td><td>81.0 ± 1.0</td><td>80.0 ± 2.0</td></tr><tr><td>Discrimination</td><td>86.0 ± 2.0</td><td>81.0 ± 5.0</td><td>85.0 ± 2.0</td><td>85.0 ± 2.0</td><td>92.8 ± 1.8</td><td>82.0 ± 4.0</td><td>82.0 ± 3.0</td></tr><tr><td>Drug abuse</td><td>93.7 ± 0.8</td><td>89.0 ± 7.0</td><td>92.0 ± 1.0</td><td>90.0 ± 2.0</td><td>90.0 ± 3.0</td><td>91.0 ± 1.0</td><td>92.0 ± 1.0</td></tr><tr><td>Financial crime</td><td>92.6 ± 1.0</td><td>92.0 ± 1.0</td><td>90.0 ± 2.0</td><td>87.0 ± 6.0</td><td>88.0 ± 1.0</td><td>91.0 ± 1.0</td><td>90.0 ± 2.0</td></tr><tr><td>Hate speech</td><td>80.0 ± 8.0</td><td>88.3 ± 0.7</td><td>82.0 ± 3.0</td><td>80.0 ± 6.0</td><td>84.0 ± 2.0</td><td>80.0 ± 9.0</td><td>83.0 ± 4.0</td></tr><tr><td>Misinformation</td><td>55.0 ± 9.0</td><td>75.5 ± 1.1</td><td>71.0 ± 7.0</td><td>72.0 ± 2.0</td><td>67.0 ± 6.0</td><td>69.0 ± 2.0</td><td>70.0 ± 2.0</td></tr><tr><td>Unethical behavior</td><td>83.0 ± 6.0</td><td>85.0 ± 1.0</td><td>85.0 ± 1.0</td><td>85.0 ± 1.0</td><td>87.5 ± 0.6</td><td>84.0 ± 1.0</td><td>81.0 ± 4.0</td></tr><tr><td>Privacy violation</td><td>89.0 ± 5.0</td><td>94.0 ± 2.0</td><td>92.0 ± 1.0</td><td>92.0 ± 2.0</td><td>93.0 ± 1.0</td><td>91.0 ± 2.0</td><td>92.0 ± 1.0</td></tr><tr><td>Self-harm</td><td>90.0 ± 3.0</td><td>93.0 ± 2.0</td><td>91.0 ± 2.0</td><td>93.0 ± 3.0</td><td>92.0 ± 3.0</td><td>92.0 ± 2.0</td><td>92.5 ± 1.6</td></tr><tr><td>Adult content</td><td>90.0 ± 3.0</td><td>93.5 ± 1.0</td><td>91.0 ± 2.0</td><td>90.0 ± 2.0</td><td>89.0 ± 2.0</td><td>89.0 ± 2.0</td><td>86.0 ± 3.0</td></tr><tr><td>Terrorism</td><td>90.0 ± 3.0</td><td>91.0 ± 2.0</td><td>89.1 ± 1.1</td><td>92.0 ± 1.0</td><td>88.0 ± 5.0</td><td>90.0 ± 3.0</td><td>85.0 ± 9.0</td></tr><tr><td>Violence</td><td>87.0 ± 2.0</td><td>87.0 ± 2.0</td><td>87.0 ± 3.0</td><td>86.0 ± 3.0</td><td>90.9 ± 0.5</td><td>86.0 ± 3.0</td><td>86.0 ± 3.0</td></tr></table>
+
+Table 15: Test accuracy (%) comparison across different numbers of facets for each category in Ministrial-8B. Results show mean ± standard deviation. Bold numbers indicate the selected configuration in our final model.  
+
+<table><tr><td>Category</td><td>1 facet</td><td>10 facets</td><td>20 facets</td><td>30 facets</td><td>40 facets</td><td>50 facets</td><td>60 facets</td></tr><tr><td>Animal abuse</td><td>50.0 ± 0.0</td><td>80.0 ± 20.0</td><td>93.6 ± 1.4</td><td>88.0 ± 6.0</td><td>80.0 ± 17.0</td><td>61.0 ± 11.0</td><td>84.0 ± 4.0</td></tr><tr><td>Child abuse</td><td>86.0 ± 6.0</td><td>89.9 ± 1.3</td><td>85.0 ± 2.0</td><td>86.0 ± 4.0</td><td>88.0 ± 1.0</td><td>88.0 ± 2.0</td><td>85.0 ± 2.0</td></tr><tr><td>Politics</td><td>86.9 ± 0.4</td><td>76.0 ± 5.0</td><td>62.0 ± 13.0</td><td>57.0 ± 8.0</td><td>76.0 ± 6.0</td><td>73.0 ± 8.0</td><td>65.0 ± 17.0</td></tr><tr><td>Discrimination</td><td>82.0 ± 2.0</td><td>76.0 ± 11.0</td><td>61.0 ± 13.0</td><td>78.0 ± 6.0</td><td>70.0 ± 9.0</td><td>87.2 ± 0.3</td><td>69.0 ± 13.0</td></tr><tr><td>Drug abuse</td><td>93.7 ± 0.8</td><td>68.0 ± 14.0</td><td>84.0 ± 14.0</td><td>73.0 ± 16.0</td><td>73.0 ± 20.0</td><td>82.0 ± 19.0</td><td>88.0 ± 7.0</td></tr><tr><td>Financial crime</td><td>92.6 ± 1.0</td><td>91.0 ± 1.0</td><td>91.0 ± 1.0</td><td>91.0 ± 1.0</td><td>90.0 ± 2.0</td><td>90.0 ± 1.0</td><td>91.0 ± 1.0</td></tr><tr><td>Hate speech</td><td>86.0 ± 4.0</td><td>88.3 ± 0.7</td><td>82.0 ± 7.0</td><td>85.0 ± 3.0</td><td>87.0 ± 2.0</td><td>81.0 ± 4.0</td><td>81.0 ± 3.0</td></tr><tr><td>Misinformation</td><td>56.0 ± 12.0</td><td>75.5 ± 1.1</td><td>73.0 ± 2.0</td><td>70.0 ± 7.0</td><td>71.0 ± 2.0</td><td>72.0 ± 3.0</td><td>68.0 ± 8.0</td></tr><tr><td>Unethical behavior</td><td>85.0 ± 2.0</td><td>85.0 ± 1.0</td><td>87.8 ± 0.6</td><td>86.0 ± 0.0</td><td>84.0 ± 1.0</td><td>82.0 ± 4.0</td><td>82.0 ± 5.0</td></tr><tr><td>Privacy violation</td><td>84.0 ± 10.0</td><td>94.0 ± 2.0</td><td>93.0 ± 2.0</td><td>91.0 ± 3.0</td><td>91.0 ± 5.0</td><td>93.0 ± 2.0</td><td>91.0 ± 4.0</td></tr><tr><td>Self-harm</td><td>91.0 ± 2.0</td><td>91.0 ± 6.0</td><td>92.5 ± 1.6</td><td>91.0 ± 2.0</td><td>92.0 ± 2.0</td><td>89.0 ± 3.0</td><td>93.0 ± 2.0</td></tr><tr><td>Adult content</td><td>92.0 ± 3.0</td><td>93.5 ± 1.0</td><td>91.0 ± 3.0</td><td>93.0 ± 2.0</td><td>93.0 ± 0.0</td><td>92.0 ± 2.0</td><td>91.0 ± 2.0</td></tr><tr><td>Terrorism</td><td>87.0 ± 5.0</td><td>89.0 ± 1.0</td><td>89.1 ± 1.1</td><td>88.0 ± 1.0</td><td>89.0 ± 1.0</td><td>88.0 ± 1.0</td><td>87.0 ± 2.0</td></tr><tr><td>Violence</td><td>88.0 ± 4.0</td><td>90.0 ± 1.0</td><td>88.0 ± 3.0</td><td>90.9 ± 0.5</td><td>87.0 ± 2.0</td><td>86.0 ± 4.0</td><td>89.0 ± 1.0</td></tr></table>
+
+Table 16: Test accuracy (%) comparison across different numbers of facets for each category in Qwen2-1.5B. Results show mean ± standard deviation. Bold numbers indicate the selected configuration in our final model.  
+
+<table><tr><td>Category</td><td>1 facet</td><td>10 facets</td><td>20 facets</td><td>30 facets</td><td>40 facets</td><td>50 facets</td><td>60 facets</td></tr><tr><td>Animal abuse</td><td>93.5 ± 1.1</td><td>89.0 ± 6.0</td><td>88.0 ± 5.0</td><td>84.0 ± 7.0</td><td>84.0 ± 7.0</td><td>86.0 ± 4.0</td><td>83.0 ± 10.0</td></tr><tr><td>Child abuse</td><td>83.0 ± 8.0</td><td>86.2 ± 2.2</td><td>85.0 ± 2.0</td><td>83.0 ± 3.0</td><td>84.0 ± 3.0</td><td>85.0 ± 2.0</td><td>82.0 ± 4.0</td></tr><tr><td>Politics</td><td>81.0 ± 3.0</td><td>80.0 ± 6.0</td><td>77.0 ± 8.0</td><td>77.0 ± 6.0</td><td>78.0 ± 2.0</td><td>83.3 ± 1.0</td><td>77.0 ± 7.0</td></tr><tr><td>Discrimination</td><td>76.0 ± 6.0</td><td>76.0 ± 5.0</td><td>78.0 ± 3.0</td><td>85.3 ± 0.4</td><td>76.0 ± 10.0</td><td>80.0 ± 4.0</td><td>81.0 ± 1.0</td></tr><tr><td>Drug abuse</td><td>90.0 ± 3.0</td><td>88.0 ± 6.0</td><td>92.6 ± 0.9</td><td>87.0 ± 8.0</td><td>90.0 ± 1.0</td><td>89.0 ± 4.0</td><td>89.0 ± 5.0</td></tr><tr><td>Financial crime</td><td>90.0 ± 3.0</td><td>88.0 ± 3.0</td><td>91.0 ± 1.0</td><td>89.0 ± 2.0</td><td>89.0 ± 4.0</td><td>89.0 ± 1.0</td><td>84.8 ± 1.2</td></tr><tr><td>Hate speech</td><td>82.0 ± 5.0</td><td>84.8 ± 1.2</td><td>82.0 ± 3.0</td><td>83.0 ± 5.0</td><td>85.0 ± 2.0</td><td>84.0 ± 1.0</td><td>85.0 ± 1.0</td></tr><tr><td>Misinformation</td><td>50.0 ± 0.0</td><td>63.0 ± 7.0</td><td>71.0 ± 2.0</td><td>71.5 ± 2.4</td><td>69.0 ± 1.0</td><td>67.0 ± 5.0</td><td>68.0 ± 5.0</td></tr><tr><td>Unethical behavior</td><td>81.0 ± 7.0</td><td>83.0 ± 3.0</td><td>86.3 ± 0.7</td><td>82.0 ± 4.0</td><td>80.0 ± 3.0</td><td>83.0 ± 1.0</td><td>82.0 ± 1.0</td></tr><tr><td>Privacy violation</td><td>90.0 ± 3.0</td><td>88.0 ± 4.0</td><td>90.0 ± 2.0</td><td>89.0 ± 2.0</td><td>93.1 ± 0.4</td><td>91.0 ± 1.0</td><td>88.0 ± 2.0</td></tr><tr><td>Self-harm</td><td>77.0 ± 14.0</td><td>86.0 ± 4.0</td><td>90.4 ± 1.6</td><td>88.0 ± 3.0</td><td>88.0 ± 2.0</td><td>88.0 ± 2.0</td><td>87.0 ± 2.0</td></tr><tr><td>Adult content</td><td>89.0 ± 1.0</td><td>91.5 ± 0.5</td><td>85.0 ± 3.0</td><td>88.0 ± 4.0</td><td>86.0 ± 2.0</td><td>88.0 ± 2.0</td><td>86.0 ± 5.0</td></tr><tr><td>Terrorism</td><td>85.0 ± 6.0</td><td>90.0 ± 1.0</td><td>88.0 ± 3.0</td><td>88.0 ± 2.0</td><td>88.0 ± 2.0</td><td>89.7 ± 1.6</td><td>89.0 ± 2.0</td></tr><tr><td>Violence</td><td>85.0 ± 3.0</td><td>81.0 ± 9.0</td><td>72.0 ± 15.0</td><td>88.2 ± 1.3</td><td>84.0 ± 2.0</td><td>86.0 ± 2.0</td><td>84.0 ± 3.0</td></tr></table>
+
+Table 17: Classification accuracy (%) on the BeaverTails dataset. Results show mean ± standard deviation across different safety categories. MLP uses an architecture with a similar amount of Safety Polytope parameters, but has additional parameters in the classification layer.  
+
+<table><tr><td rowspan="2">Category</td><td colspan="3">Llama2-7B</td><td colspan="3">Ministral-8B</td><td colspan="3">Qwen2-1.5B</td></tr><tr><td>1 facet</td><td>SaP</td><td>MLP</td><td>1 facet</td><td>SaP</td><td>MLP</td><td>1 facet</td><td>SaP</td><td>MLP</td></tr><tr><td>Animal abuse</td><td>60.0 ± 18.0</td><td>93.0 ± 1.0</td><td>96.0</td><td>50.0 ± 0.0</td><td>93.6 ± 1.4</td><td>95.5</td><td>93.5 ± 1.1</td><td>93.5 ± 1.1</td><td>94.8</td></tr><tr><td>Child abuse</td><td>85.0 ± 8.0</td><td>91.8 ± 2.2</td><td>92.8</td><td>86.0 ± 6.0</td><td>89.9 ± 1.3</td><td>91.7</td><td>83.0 ± 8.0</td><td>86.2 ± 2.2</td><td>87.6</td></tr><tr><td>Politics</td><td>82.0 ± 3.0</td><td>84.1 ± 0.5</td><td>84.1</td><td>86.9 ± 0.4</td><td>86.9 ± 0.4</td><td>86.8</td><td>81.0 ± 3.0</td><td>83.3 ± 1.0</td><td>85.1</td></tr><tr><td>Discrimination</td><td>86.0 ± 2.0</td><td>92.8 ± 1.8</td><td>88.9</td><td>82.0 ± 2.0</td><td>87.2 ± 0.3</td><td>88.4</td><td>76.0 ± 6.0</td><td>85.3 ± 0.4</td><td>87.0</td></tr><tr><td>Drug abuse</td><td>93.7 ± 0.8</td><td>93.7 ± 0.8</td><td>95.1</td><td>93.7 ± 0.8</td><td>93.7 ± 0.8</td><td>95.9</td><td>90.0 ± 3.0</td><td>92.6 ± 0.9</td><td>93.6</td></tr><tr><td>Financial crime</td><td>92.6 ± 1.0</td><td>92.6 ± 1.0</td><td>94.0</td><td>92.6 ± 1.0</td><td>92.6 ± 1.0</td><td>94.7</td><td>90.0 ± 3.0</td><td>91.0 ± 1.0</td><td>93.5</td></tr><tr><td>Hate speech</td><td>80.0 ± 8.0</td><td>88.3 ± 0.7</td><td>88.5</td><td>86.0 ± 4.0</td><td>88.3 ± 0.7</td><td>88.3</td><td>82.0 ± 5.0</td><td>85.0 ± 2.0</td><td>86.9</td></tr><tr><td>Misinformation</td><td>55.0 ± 9.0</td><td>75.5 ± 1.1</td><td>71.7</td><td>56.0 ± 12.0</td><td>75.5 ± 1.1</td><td>77.6</td><td>50.0 ± 0.0</td><td>71.5 ± 2.4</td><td>74.5</td></tr><tr><td>Unethical behavior</td><td>83.0 ± 6.0</td><td>87.5 ± 0.6</td><td>88.8</td><td>85.0 ± 2.0</td><td>87.8 ± 0.6</td><td>88.3</td><td>81.0 ± 7.0</td><td>86.3 ± 0.7</td><td>86.8</td></tr><tr><td>Privacy violation</td><td>89.0 ± 5.0</td><td>94.0 ± 2.0</td><td>94.9</td><td>84.0 ± 10.0</td><td>94.0 ± 2.0</td><td>96.2</td><td>90.0 ± 3.0</td><td>93.1 ± 0.4</td><td>94.7</td></tr><tr><td>Self-harm</td><td>90.0 ± 3.0</td><td>93.0 ± 2.0</td><td>92.6</td><td>91.0 ± 2.0</td><td>93.0 ± 2.0</td><td>94.3</td><td>77.0 ± 14.0</td><td>90.4 ± 1.6</td><td>89.7</td></tr><tr><td>Adult content</td><td>90.0 ± 3.0</td><td>93.5 ± 1.0</td><td>92.7</td><td>92.0 ± 3.0</td><td>93.5 ± 1.0</td><td>94.1</td><td>89.0 ± 1.0</td><td>91.5 ± 0.5</td><td>91.3</td></tr><tr><td>Terrorism</td><td>90.0 ± 3.0</td><td>92.0 ± 1.0</td><td>92.7</td><td>87.0 ± 5.0</td><td>90.0 ± 1.0</td><td>90.0</td><td>85.0 ± 6.0</td><td>90.0 ± 1.0</td><td>92.3</td></tr><tr><td>Violence</td><td>87.0 ± 2.0</td><td>90.9 ± 0.5</td><td>91.6</td><td>88.0 ± 4.0</td><td>90.9 ± 0.5</td><td>91.9</td><td>85.0 ± 3.0</td><td>88.2 ± 1.3</td><td>90.6</td></tr><tr><td>All data</td><td>81.2 ± 0.2</td><td>82.2 ± 0.4</td><td>83.9</td><td>80.2 ± 0.5</td><td>82.2 ± 0.8</td><td>83.9</td><td>79.5 ± 0.8</td><td>80.3 ± 1.0</td><td>83.1</td></tr></table>
+
+# E. Polytope Sample Complexity
+
+We present the current theoretical results of learning a polytope. Note that these results make 3 important assumptions: (i) reliance on certain restricted encoder nature; (ii) perfect data separability by the polytope; (iii) the feasibility of finding a solution.
+
+The following theorem is derived from (Gottlieb et al., 2021), which establishes the fat-shattering dimension of  $\gamma$ -fat  $t$ -polyhedra.
+
+Theorem E.1 (Theorem 2.2, Gottlieb et al. (2021)). The class of  $\gamma$ -fat  $t$ -polyhedra in  $\mathbb{R}^d$ , with at most  $t$  hyperplanes, has a fat-shattering dimension of order:
+
+$$
+D = O \left(\frac {t \log t}{\gamma^ {2}}\right),
+$$
+
+where  $t$  is the number of hyperplanes defining the polyhedron, and  $\gamma$  is the margin.
+
+This result provides the fat-shattering dimension  $D$  for our hypothesis class (the class of  $\gamma$ -fat polytopes), which we will use to derive the sample complexity.
+
+The second theorem we rely on is a general result from Anthony & Bartlett (1999), which gives a sample complexity bound based on the fat-shattering dimension.
+
+Theorem E.2 (Theorem 19.1, Anthony & Bartlett (1999)). Suppose that  $F$  is a class of functions mapping from a domain  $X$  into the real interval [0, 1], and suppose also that  $F$  has finite fat-shattering dimension. Let  $\mathcal{A}$  be any approximate-SEM algorithm for  $F$  and define, for  $z \in Z^m$ ,  $L(z) = \mathcal{A}(z, \epsilon_0 / 6)$ , where  $\epsilon_0 = 16 / \sqrt{m}$ . Then  $L$  is a learning algorithm for  $F$ , and its sample complexity satisfies
+
+$$
+m (\epsilon , \delta) \leq \frac {2 5 6}{\epsilon^ {2}} \left(1 8 \mathrm {f a t} _ {F} \left(\frac {\epsilon}{2 5 6}\right) \ln^ {2} \left(\frac {1 2 8}{\epsilon}\right) + \ln \left(\frac {1 6}{\delta}\right)\right)
+$$
+
+for all  $\epsilon, \delta > 0$ .
+
+Using big-O notation, we simplify this to the following form:
+
+$$
+m (\epsilon , \delta) = O \left(\frac {1}{\epsilon^ {2}} \left(\operatorname {f a t} _ {F} (\epsilon) \log^ {2} \left(\frac {1}{\epsilon}\right) + \log \left(\frac {1}{\delta}\right)\right)\right).
+$$
+
+We now state and prove the main result, which gives the sample complexity bound for learning  $\gamma$ -fat polytopes.
+
+Corollary E.3 (Sample complexity of  $\gamma$ -fat polytopes). Let  $\mathcal{H}$  be the class of  $\gamma$ -fat  $t$ -polyhedra in  $\mathbb{R}^d$ . The sample complexity  $m(\epsilon, \delta)$  required to learn a polytope from  $\mathcal{H}$  with accuracy  $\epsilon$  and confidence  $1 - \delta$  is bounded by:
+
+$$
+m (\epsilon , \delta) = O \left(\frac {1}{\epsilon^ {2}} \left(\frac {t \log t}{\gamma^ {2}} \log^ {2} \left(\frac {1}{\epsilon}\right) + \log \left(\frac {1}{\delta}\right)\right)\right).
+$$
+
+Proof. By Theorem E.1, the fat-shattering dimension of the class of  $\gamma$ -fat  $t$ -polyhedra is:
+
+$$
+D = O \left(\frac {t \log t}{\gamma^ {2}}\right).
+$$
+
+Substituting this into the sample complexity bound from Theorem E.2, we get:
+
+$$
+m (\epsilon , \delta) = O \left(\frac {1}{\epsilon^ {2}} \left(\frac {t \log t}{\gamma^ {2}} \log^ {2} \left(\frac {1}{\epsilon}\right) + \log \left(\frac {1}{\delta}\right)\right)\right).
+$$
+
+Thus, the sample complexity of learning a  $\gamma$ -fat polytope is as stated.

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+# Learning Single Index Models with Diffusion Priors
+
+Anqi Tang *1 Youming Chen *1 Shuchen Xue23 Zhaoqiang Liu1
+
+# Abstract
+
+Diffusion models (DMs) have demonstrated remarkable ability to generate diverse and high-quality images by efficiently modeling complex data distributions. They have also been explored as powerful generative priors for signal recovery, resulting in a substantial improvement in the quality of reconstructed signals. However, existing research on signal recovery with diffusion models either focuses on specific reconstruction problems or is unable to handle nonlinear measurement models with discontinuous or unknown link functions. In this work, we focus on using DMs to achieve accurate recovery from semi-parametric single index models, which encompass a variety of popular nonlinear models that may have discontinuous and unknown link functions. We propose an efficient reconstruction method that only requires one round of unconditional sampling and (partial) inversion of DMs. Theoretical analysis on the effectiveness of the proposed methods has been established under appropriate conditions. We perform numerical experiments on image datasets for different nonlinear measurement models. We observe that compared to competing methods, our approach can yield more accurate reconstructions while utilizing significantly fewer neural function evaluations.
+
+# 1. Introduction
+
+The objective of compressed sensing is to accurately recover the underlying high-dimensional sparse signal from a small number of linear measurements, with the measurement model as follows (Candes & Wakin, 2008; Foucart &
+
+*Equal contribution  $^{1}$ University of Electronic Science and Technology of China  $^{2}$ University of Chinese Academy of Sciences  $^{3}$ Academy of Mathematics and Systems Science, CAS. Correspondence to: Zhaoqiang Liu <zqliu12@gmail.com>.
+
+Proceedings of the  $42^{nd}$  International Conference on Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).
+
+Rauhut, 2013; Scarlett & Cevher, 2021):
+
+$$
+\boldsymbol {y} = \mathbf {A} \boldsymbol {x} ^ {*} + \boldsymbol {e}, \tag {1}
+$$
+
+where  $\mathbf{A} \in \mathbb{R}^{m \times n}$  is the measurement matrix,  $\pmb{x}^* \in \mathbb{R}^n$  is the signal to be recovered,  $\pmb{e} \in \mathbb{R}^m$  is the additive noise vector, and  $\pmb{y} \in \mathbb{R}^m$  is the observed vector.
+
+Although the linear measurement model utilized in compressed sensing can serve as a good testbed for demonstrating conceptual phenomena, in many real-world problems, it may not be justifiable or even feasible. For instance, the binary measurement model employed in 1-bit compressed sensing (Boufounos & Baraniuk, 2008) has attracted substantial interest. This is due to its cost-effective and efficient hardware. It has also been shown that 1-bit compressed sensing is robust against certain nonlinear distortions (Laska & Baraniuk, 2012). The limitations of the linear data model have prompted the study of general nonlinear measurement models. Among these, the semi-parametric single index model (SIM) is arguably the most popular (Horowitz, 2009). The SIM models the observations as follows:
+
+$$
+\boldsymbol {y} = f \left(\mathbf {A} \boldsymbol {x} ^ {*}\right), \tag {2}
+$$
+
+where  $\mathbf{A}$  consists of i.i.d. standard Gaussian entries, and  $f: \mathbb{R} \to \mathbb{R}$  is an unknown and possibly random nonlinear link function that is applied element-wise. The aim is to estimate the signal  $\boldsymbol{x}^*$  using the knowledge of  $\mathbf{A}$  and  $\boldsymbol{y}$ , despite the unknown nonlinear link  $f$ . It is well-known that in the SIM,  $\boldsymbol{x}^*$  is generally not identifiable (Plan & Vershynin, 2016), since any scaling of  $\boldsymbol{x}^*$  can be incorporated into the unknown  $f$ . Consequently, it is common to impose the identifiability constraint  $\| \boldsymbol{x}^* \|_2 = 1$ .
+
+To enable efficient and reliable recovery of the underlying signal  $\boldsymbol{x}^*$ , the predominant approach is to introduce structural modeling assumptions, such as sparse and conventional generative priors. For instance, the research works following (Bora et al., 2017) typically achieve signal reconstruction under the assumption that the underlying signal lies within the range of a variational autoencoder (VAE) or a generative adversarial network (GAN) (Van Veen et al., 2018; Hand et al., 2018; Heckel & Hand, 2019; Wu et al., 2019; Asim et al., 2020; Ongie et al., 2020; Whang et al., 2020; Jalal et al., 2021b; Nguyen et al., 2021; Liu et al., 2021b; 2022a; Genzel et al., 2022; Liu et al., 2024).
+
+Compared to conventional generative models like VAEs and GANs, diffusion models (DMs) (Sohl-Dickstein et al., 2015; Song & Ermon, 2019; Ho et al., 2020) have recently demonstrated their superiority in terms of robust generative capabilities and learning complex data distributions (Dhariwal & Nichol, 2021). DMs have been integrated into a wide variety of domains, leading to transformative progress and reshaping traditional approaches in areas such as vision and audio generation (Saharia et al., 2022b; Kong et al., 2021), and reinforcement learning (Chi et al., 2023; Hansen-Estruch et al., 2023). In particular, when it comes to addressing signal recovery problems, diffusion models offer a more flexible and robust solution, facilitating high-quality reconstructions and improved performance (Kawar et al., 2022; Whang et al., 2022; Sahara et al., 2022c;a; Alkan et al., 2023; Chung et al., 2023a; Cardoso et al., 2023; Wang et al., 2023; Chung et al., 2023b; Feng & Bouman, 2023; Rout et al., 2023; Wu et al., 2023; Aali et al., 2024; Chung et al., 2024; Dou & Song, 2024; Song et al., 2024; Wu et al., 2024; Sun et al., 2024).
+
+Inspired by the strong capabilities of DMs in diverse real-world applications, in this study, we investigate the application of DMs to learn SIMs.
+
+# 1.1. Related Work
+
+Relevant works on signal recovery with conventional generative models (such as GANs and VAEs) have shown that learned priors can drastically reduce sample complexity in compressed sensing and related problems. A comprehensive discussion of diffusion-based and conventional generative approaches is provided in Appendix A. In the following, we highlight diffusion-based signal recovery methods.
+
+Signal recovery with diffusion models: Diffusion models (DMs) have advanced signal recovery by modeling complex data distributions, with two main paradigms: task-specific and pre-trained approaches. Task-specific DMs are trained for particular problems, such as super-resolution (Saharia et al., 2022c) or deblurring (Ren et al., 2023). Pre-trained DMs, used without task-specific training, guide reconstruction via score estimation (e.g., (Feng et al., 2023)) or adapt frameworks like DDPM for linear measurements (e.g., DDRM (Kawar et al., 2022)). MCG (Chung et al., 2022a) and IIGDM (Song et al., 2023) are mainly designed for the linear setting. Additionally, IIGDM can be extended to certain nonlinear settings (where the link function is known) by leveraging a combination of pseudoinverse operations and nonlinear transformations. DPS (Chung et al., 2023b) and DAPS (Zhang et al., 2024a) are applicable in nonlinear settings as well, also primarily under the assumption that the link function is known.
+
+However, the works mentioned above either concentrate on
+
+specific linear signal recovery problems or are incapable of handling nonlinear measurement models with discontinuous or unknown link functions. Perhaps most relevant to the present paper, the work (Meng & Kabashima, 2022) uses SGMs to learn the underlying distribution of natural signals. By estimating the conditional score function, it enables better reconstruction quality when dealing with quantized compressed sensing problems. Nevertheless, the approach in (Meng & Kabashima, 2022) is restricted to quantized measurements and is relatively slow in reconstruction. Even if utilizing the knowledge of the link function, it requires thousands of neural function evaluations (NFEs) to achieve a reasonably accurate recovery.
+
+# 1.2. Contributions
+
+The main contributions of this paper are threefold:
+
+- We propose an efficient method for accurately recovering the underlying signal from nonlinear observations of SIMs by utilizing diffusion priors. Our approach does not require the knowledge of the link function, and it only requires one round of unconditional sampling and (partial) inversion of DMs, resulting in a relatively small number of NFEs for the entire process.  
+- Based on heuristic theoretical results, we opt to initiate the inversion of DMs from an intermediate time rather than performing the full inversion of DMs. Under suitable conditions, we provide a theoretical analysis on the effectiveness of the corresponding approach. Additionally, we empirically observe that carrying out partial inversion results in significantly better reconstructions compared to performing full inversion.  
+- To validate the effectiveness of our method, we conduct numerical experiments on image datasets under distinct nonlinear measurement models. Notably, in the noisy 1-bit measurement setting, our approach not only achieves substantially faster computational performance but also outperforms competing methods in reconstruction accuracy, even when these methods explicitly incorporate knowledge of the link function.
+
+# 1.3. Notation
+
+We use upper and lower case boldface letters to denote matrices and vectors respectively. For any  $M \in \mathbb{N}$ , we use the shorthand notation  $[M] = \{1, 2, \dots, M\}$ , and we use  $\pmb{I}_M$  to denote the identity matrix in  $\mathbb{R}^{M \times M}$ . Given two sequences of real values  $\{a_i\}$  and  $\{b_i\}$ , we write  $a_i = O(b_i)$  if there exists an absolute constant  $C_1$  and a positive integer  $i_1$  such that for any  $i > i_1$ ,  $|a_i| \leq C_1 b_i$ . We use  $\mathcal{S}^{n-1}$  to denote the unit sphere in  $\mathbb{R}^n$ , i.e.,  $\mathcal{S}^{n-1} = \{s \in \mathbb{R}^n : \|s\|_2 = 1\}$ . For a generator  $G: \mathbb{R}^n \to \mathbb{R}^n$ , we use  $\mathcal{R}(G)$  to denote the range of  $G$ , i.e.,  $\mathcal{R}(G) = \{G(z): z \in \mathbb{R}^n\}$ .
+
+# 2. Preliminaries for Diffusion Models
+
+Diffusion Models (DMs) define a forward process  $\{\pmb{x}_t\}_{t\in [0,T]}$  starting from  $t = 0$  and  $\pmb{x}_0$ . For any  $t\in [0,T]$ , the distribution of  $\pmb{x}_t$  conditioned on  $\pmb{x}_0$  satisfies the following equation:
+
+$$
+q _ {0 t} \left(\boldsymbol {x} _ {t} \mid \boldsymbol {x} _ {0}\right) = \mathcal {N} \left(\alpha_ {t} \boldsymbol {x} _ {0}, \sigma_ {t} ^ {2} \boldsymbol {I} _ {n}\right). \tag {3}
+$$
+
+Here,  $\alpha_{t}$  and  $\sigma_{t}$  are non-negative, differentiable functions possessing bounded derivatives. Moreover,  $\alpha_{t}$  is monotonically non-increasing with  $\alpha_0 = 1$ , while  $\sigma_{t}$  is monotonically increasing. It has been demonstrated that the following stochastic differential equation (SDE) has the same transition distribution  $q_{0t}(\pmb{x}_t|\pmb{x}_0)$  for any  $t\in [0,T]$  (Song et al., 2021b; Kingma et al., 2021):
+
+$$
+\mathrm {d} \boldsymbol {x} _ {t} = f (t) \boldsymbol {x} _ {t} \mathrm {d} t + g (t) \mathrm {d} \boldsymbol {w} _ {t}, \quad \boldsymbol {x} _ {0} \sim q _ {0} (\boldsymbol {x} _ {0}), \tag {4}
+$$
+
+where  $\pmb{w}_t \in \mathbb{R}^n$  is the standard Wiener process, and  $q_t$  represents the marginal distribution of  $\pmb{x}_t$ . Moreover,  $f(t)$  is a scalar function known as the drift coefficient and  $g(t)$  is a scalar function called the diffusion coefficient, which are specified as follows:
+
+$$
+f (t) = \frac {\mathrm {d} \log \alpha_ {t}}{\mathrm {d} t}, g ^ {2} (t) = \frac {\mathrm {d} \sigma_ {t} ^ {2}}{\mathrm {d} t} - 2 \frac {\mathrm {d} \log \alpha_ {t}}{\mathrm {d} t} \sigma_ {t} ^ {2}. (5)
+$$
+
+Under certain regularity conditions, Song et al. (2021b) demonstrate that the forward process in Eq. (4) has an equivalent reverse process running from time  $T$  to 0, which starts with the marginal distribution  $q_{T}(\pmb{x}_{T})$ :
+
+$$
+\mathrm {d} \boldsymbol {x} _ {t} = [ f (t) \boldsymbol {x} _ {t} - g ^ {2} (t) \nabla_ {\boldsymbol {x}} \log q _ {t} (\boldsymbol {x} _ {t}) ] \mathrm {d} t + g (t) \mathrm {d} \bar {\boldsymbol {w}} _ {t}, \tag {6}
+$$
+
+where  $\bar{\boldsymbol{w}}_t\in \mathbb{R}^n$  represents the Wiener process in reverse time, and  $\nabla_{\pmb{x}}\log q_t(\pmb {x}_t)$  stands for the time-varying score function.
+
+For more efficient sampling, Song et al. (2021b) further show that the following probability flow ordinary differential equation (ODE) possesses the same marginal distribution at each time  $t$  as that of reverse SDE in Eq. (6):
+
+$$
+\frac {\mathrm {d} \boldsymbol {x} _ {t}}{\mathrm {d} t} = f (t) \boldsymbol {x} _ {t} - \frac {1}{2} g ^ {2} (t) \nabla_ {\boldsymbol {x}} \log q _ {t} (\boldsymbol {x} _ {t}), \quad \boldsymbol {x} _ {T} \sim q _ {T} (\boldsymbol {x} _ {T}). \tag {7}
+$$
+
+The only unknown term in Eq. (7) is the time-varying score function  $\nabla_{\pmb{x}}\log q_t(\pmb{x}_t)$ . In practice, DMs typically estimate the scaled score function  $-\sigma_t\nabla_{\pmb{x}}\log q_t(\pmb{x}_t)$  by using a noise prediction network  $\epsilon_{\theta}(\pmb{x}_t,t)$ , and optimizes the parameter  $\pmb{\theta}$  through minimizing the following objective:
+
+$$
+\int_ {0} ^ {T} \mathbb {E} _ {\boldsymbol {x} _ {0} \sim q _ {0}} \mathbb {E} _ {\boldsymbol {\epsilon} \sim \mathcal {N} (\boldsymbol {0}, \mathbf {I} _ {n})} \left[ \| \boldsymbol {\epsilon} _ {\boldsymbol {\theta}} \left(\alpha_ {t} \boldsymbol {x} _ {0} + \sigma_ {t} \boldsymbol {\epsilon}, t\right) - \boldsymbol {\epsilon} \| _ {2} ^ {2} \right] d t. \tag {8}
+$$
+
+By substituting  $\nabla_{\pmb{x}}\log p_t(\pmb{x}_t) = -\epsilon_\pmb{\theta}(\pmb{x}_t,t) / \sigma_t$  into Eq. (7) and taking advantage of its semi-linear structure, we obtain the following numerical integration formula (Lu et al., 2022a; Zhang & Chen, 2022):
+
+$$
+\boldsymbol {x} _ {t} = e ^ {\int_ {s} ^ {t} f (\tau) \mathrm {d} \tau} \boldsymbol {x} _ {s} + \int_ {s} ^ {t} \left(e ^ {\int_ {\tau} ^ {t} f (r) \mathrm {d} r} \cdot \frac {g ^ {2} (\tau)}{2 \sigma_ {\tau}} \cdot \boldsymbol {\epsilon} _ {\theta} \left(\boldsymbol {x} _ {\tau}, \tau\right)\right) \mathrm {d} \tau . \tag {9}
+$$
+
+Alternatively, we can use a data prediction network  $\pmb{x}_{\theta}(\pmb{x}_t,t)$  that satisfies the condition  $\epsilon_{\theta}(\pmb{x}_t,t) = (\pmb{x}_t - \alpha_t\pmb{x}_{\theta}(\pmb{x}_t,t)) / \sigma_t$  (Kingma et al., 2021; Salimans & Ho, 2022). By substituting Eq. (5) into Eq. (9) and using the transition between  $\epsilon_{\theta}$  and  $\pmb{x}_{\theta}$ , we obtain the following integration formula (Lu et al., 2022b):
+
+$$
+\boldsymbol {x} _ {t} = \frac {\sigma_ {t}}{\sigma_ {s}} \boldsymbol {x} _ {s} + \sigma_ {t} \int_ {\lambda_ {s}} ^ {\lambda_ {t}} e ^ {\lambda} \hat {\boldsymbol {x}} _ {\boldsymbol {\theta}} (\hat {\boldsymbol {x}} _ {\lambda}, \lambda) \mathrm {d} \lambda , \tag {10}
+$$
+
+where  $\lambda_{t} := \log (\alpha_{t} / \sigma_{t})$  is strictly decreasing with respect to  $t$  and has an inverse function  $t_{\lambda}(\cdot)$ . Thus,  $\pmb{x}_{\theta}(\pmb{x}_{t}, t)$  can be written as  $\pmb{x}_{\theta}\big(\pmb{x}_{t_{\lambda}(\lambda)}, t_{\lambda}(\lambda)\big) = \hat{\pmb{x}}_{\theta}(\hat{\pmb{x}}_{\lambda}, \lambda)$ .
+
+# 2.1. The Sampling Process of Diffusion Models
+
+The sampling process in DMs gradually denoises a noise vector to an image vector that approximately follows the same distribution as the training data. More precisely, if we divide the time interval  $[\epsilon, T]^1$  into  $N$  sub-intervals with  $\epsilon = t_N < t_{N-1} < \ldots < t_1 < t_0 = T$ , and use a first-order numerical scheme for Eq. (10), for  $i \in [N]$ , we can derive the following iterative formula for the transition from time  $t_{i-1}$  to  $t_i$ :
+
+$$
+\tilde {\boldsymbol {x}} _ {t _ {i}} = \frac {\sigma_ {t _ {i}}}{\sigma_ {t _ {i - 1}}} \tilde {\boldsymbol {x}} _ {t _ {i - 1}} + \sigma_ {t _ {i}} \left(\frac {\alpha_ {t _ {i}}}{\sigma_ {t _ {i}}} - \frac {\alpha_ {t _ {i - 1}}}{\sigma_ {t _ {i - 1}}}\right) \cdot \boldsymbol {x} _ {\boldsymbol {\theta}} \left(\tilde {\boldsymbol {x}} _ {t _ {i - 1}}, t _ {i - 1}\right), \tag {11}
+$$
+
+which is in line with the widely-used DDIM sampling method (Song et al., 2021a).
+
+Moreover, second- or third-order numerical schemes can be employed to enhance the sampling efficiency further (Lu et al., 2022b; Zhao et al., 2024). Specifically, the second-order multi-step numerical solver for Eq. (10) yields the following iterative formula for  $i \geq 2$  (Lu et al., 2022b):
+
+$$
+\begin{array}{l} \tilde {\boldsymbol {x}} _ {t _ {i}} = \frac {\sigma_ {t _ {i}}}{\sigma_ {t _ {i - 1}}} \tilde {\boldsymbol {x}} _ {t _ {i - 1}} - \alpha_ {t _ {i}} (e ^ {- h _ {i}} - 1) \\ \times \left(\left(1 + \frac {1}{2 r _ {i}}\right) \boldsymbol {x} _ {\boldsymbol {\theta}} \left(\tilde {\boldsymbol {x}} _ {t _ {i - 1}}, t _ {i - 1}\right) - \frac {1}{2 r _ {i}} \boldsymbol {x} _ {\boldsymbol {\theta}} \left(\tilde {\boldsymbol {x}} _ {t _ {i - 2}}, t _ {i - 2}\right)\right), \tag {12} \\ \end{array}
+$$
+
+where  $h_i = \lambda_{t_i} - \lambda_{t_{i-1}}$  and  $r_i = h_{i-1} / h_i$ . We refer to the corresponding sampling method as DM2M for brevity.
+
+Let  $\kappa_{i}$  be the function corresponding to the  $i$ -th sampling step. For instance, for the DDIM sampling method in Eq. (11), we have
+
+$$
+\kappa_ {i} (\boldsymbol {x}) := \frac {\sigma_ {t _ {i}}}{\sigma_ {t _ {i - 1}}} \boldsymbol {x} + \sigma_ {t _ {i}} \left(\frac {\alpha_ {t _ {i}}}{\sigma_ {t _ {i}}} - \frac {\alpha_ {t _ {i - 1}}}{\sigma_ {t _ {i - 1}}}\right) \cdot \boldsymbol {x} _ {\boldsymbol {\theta}} (\boldsymbol {x}, t _ {i - 1}). \tag {13}
+$$
+
+Then, the entire sampling process can be expressed as  $\tilde{\pmb{x}}_{\epsilon} = G(\tilde{\pmb{x}}_T)$ , where the generator  $G:\mathbb{R}^n\to \mathbb{R}^n$  is the composition of of  $\kappa_{1},\kappa_{2},\ldots ,\kappa_{N}$ , i.e.,
+
+$$
+G (\boldsymbol {x}) = \kappa_ {N} \circ \dots \circ \kappa_ {2} \circ \kappa_ {1} (\boldsymbol {x}). \tag {14}
+$$
+
+In addition, for convenience, for any  $t \in (\epsilon, T]$ , we use  $G_t$  to denote the sampling process from  $t$  to  $\epsilon$  (and thus  $G_T = G$ ). Specifically, if letting  $i_t = \max \{i \in [N] : t_{i-1} \geq t\}$ , we have that  $G_t$  is the composition of  $\kappa_{i_t}, \kappa_{i_t+1}, \ldots, \kappa_N$ , i.e.,
+
+$$
+G _ {t} (\boldsymbol {x}) = \kappa_ {N} \circ \dots \circ \kappa_ {i _ {t} + 1} \circ \kappa_ {i _ {t}} (\boldsymbol {x}). \tag {15}
+$$
+
+# 2.2. The Inversion of Diffusion Models
+
+The inversion in DMs retraces the ODE sampling process with the aim of identifying the specific noise vector that generates a given image vector. This inversion is a crucial aspect in various applications, including image editing (Hertz et al., 2023; Kim et al., 2022; Wallace et al., 2023; Patashnik et al., 2023), watermark detection (Wen et al., 2023), and imaged-to-image translation (Kawar et al., 2022; Su et al., 2023). Recently, the inversion of DMs has attracted increasing attention, and several relevant methods have emerged (Pan et al., 2023; Zhang et al., 2024b; Wallace et al., 2023; Hong et al., 2024; Wang et al., 2024).
+
+For example, the naive DDIM inversion method (Hertz et al., 2023; Kim et al., 2022) has the following iterative procedure:
+
+$$
+\hat {\boldsymbol {x}} _ {t _ {i - 1}} = \frac {\sigma_ {t _ {i - 1}}}{\sigma_ {t _ {i}}} \hat {\boldsymbol {x}} _ {t _ {i}} + \sigma_ {t _ {i - 1}} \left(\frac {\alpha_ {t _ {i - 1}}}{\sigma_ {t _ {i - 1}}} - \frac {\alpha_ {t _ {i}}}{\sigma_ {t _ {i}}}\right) \boldsymbol {x} _ {\boldsymbol {\theta}} \left(\hat {\boldsymbol {x}} _ {t _ {i}}, t _ {i - 1}\right). \tag {16}
+$$
+
+The naive DDIM inversion method circumvents the computational overhead of the implicit numerical scheme by substituting  $\boldsymbol{x}_{\theta}(\hat{\boldsymbol{x}}_{t_{i - 1}}, t_{i - 1})$  with  $\boldsymbol{x}_{\theta}(\hat{\boldsymbol{x}}_{t_i}, t_{i - 1})$ . It has been widely used in imaging applications like image editing. Alternatively, by taking into account the first-order numerical discretization of Eq. (10) from  $t = \epsilon$  to  $t = T$ , it is straightforward to derive an iterative formula that is slightly
+
+different from Eq. (16):
+
+$$
+\hat {\boldsymbol {x}} _ {t _ {i - 1}} = \frac {\sigma_ {t _ {i - 1}}}{\sigma_ {t _ {i}}} \hat {\boldsymbol {x}} _ {t _ {i}} + \sigma_ {t _ {i - 1}} \left(\frac {\alpha_ {t _ {i - 1}}}{\sigma_ {t _ {i - 1}}} - \frac {\alpha_ {t _ {i}}}{\sigma_ {t _ {i}}}\right) \boldsymbol {x} _ {\boldsymbol {\theta}} \left(\hat {\boldsymbol {x}} _ {t _ {i}}, t _ {i}\right). \tag {17}
+$$
+
+Furthermore, similar to DM2M in Eq. (12), by considering the second-order numerical discretization of Eq. (10) from  $t = \epsilon$  to  $t = T$ , we can obtain the corresponding iterative formula for the inversion of DMs.
+
+Letting  $v_{i}$  be the function corresponding to the  $i$ -th inversion step. Then, the entire inversion procedure can be written as  $\hat{\pmb{x}}_T = G^\dagger (\hat{\pmb{x}}_\epsilon)$ , where the inversion operator  $G^{\dagger}:\mathbb{R}^{n}\to \mathbb{R}^{n}$  is the composition of  $v_{1},v_{2},\ldots ,v_{N}$ , i.e.,
+
+$$
+G ^ {\dagger} (\boldsymbol {x}) = v _ {1} \circ v _ {2} \circ \dots \circ v _ {N} (\boldsymbol {x}). \tag {18}
+$$
+
+In addition, for convenience, for any  $t \in [\epsilon, T)$ , we use  $G_t^\dagger$  to denote the inversion from  $t$  to  $T$  (and thus  $G_\epsilon^\dagger = G^\dagger$ ). Specifically, if letting  $j_t = \min \{j \in [M] : t_j \leq t\}$ , we have that  $G_t^\dagger$  is the composition of  $v_{j_t}, v_{j_{t-1}}, \ldots, v_1$ , i.e.,
+
+$$
+G _ {t} ^ {\dagger} (\boldsymbol {x}) = v _ {1} \circ v _ {2} \circ \dots \circ v _ {j _ {t}} (\boldsymbol {x}). \tag {19}
+$$
+
+# 3. Setup and Approaches
+
+Recall that the nonlinear observation vector  $\pmb{y} \in \mathbb{R}^{m}$  is assumed to be generated in accordance with the SIM in Eq. (2), where  $\pmb{x}^{*} \in S^{n-1}$  is the signal to be estimated. We follow (Liu & Liu, 2022) to assume that the unknown link function  $f$  satisfies the following conditions:
+
+$$
+\mu := \mathbb {E} _ {\boldsymbol {a} \sim \mathcal {N} (\boldsymbol {0}, \mathbf {I} _ {n})} \left[ f \left(\boldsymbol {a} ^ {T} \boldsymbol {x} ^ {*}\right) \boldsymbol {a} ^ {T} \boldsymbol {x} ^ {*} \right] \neq 0, \tag {20}
+$$
+
+$$
+\mathbb {E} _ {\boldsymbol {a} \sim \mathcal {N} (\boldsymbol {0}, \mathbf {I} _ {n})} \left[ f \left(\boldsymbol {a} ^ {T} \boldsymbol {x} ^ {*}\right) ^ {4} \right] <   \infty . \tag {21}
+$$
+
+The conditions in Eqs. (20) and (21) are mild and hold true for various forms of  $f$ , such as  $f(x) = \mathrm{sign}(x)$  for 1-bit measurements and  $f(x) = x^3$  for cubic measurements. However, as noted in prior works such as (Liu & Scarlett, 2020a; Liu & Liu, 2022), the condition in Eq. (20) is not satisfied for phase retrieval models with  $f(x) = x^2$  or  $f(x) = |x|$  (or their noisy versions).
+
+Remark 1. The condition in Eq. (20) is a classic and crucial condition for SIMs. For example, it is (albeit implicitly) assumed in the seminal work (Plan & Vershynin, 2016) and in subsequent research that builds upon it. If this condition fails to hold, specifically when  $\mu = 0$ , the recovery of  $\mu x^{*}$  as (Plan & Vershynin, 2016) and in our Eq. (24) below becomes meaningless. Additionally, we follow (Liu & Liu, 2022) to assume the condition in Eq. (21), which generalizes the assumption that  $f(\boldsymbol{a}^T\boldsymbol{x}^*)$  is sub-Gaussian (which is satisfied by quantized measurement models), and accommodates more general nonlinear measurement models such as cubic measurements with  $f(x) = x^3$  and their noisy counterparts.
+
+We utilize a generator  $G: \mathbb{R}^n \to \mathbb{R}^n$  that corresponds to the sampling process of DMs (cf. Eq. (14)) to model the underlying signal. Specifically, we assume that  $\pmb{x}^{*} \in \mathcal{R}(G)$ , where  $\mathcal{R}(G)$  denotes the range of  $G$ . This assumption is standard for nonlinear measurement models with generative priors and has also been made in prior works such as (Wei et al., 2019; Liu & Scarlett, 2020a).
+
+Let  $G^{\dagger}:\mathbb{R}^{n}\to \mathbb{R}^{n}$  be an operator corresponding to an inversion method for DMs (cf. Eq. (18)). The work (Liu & Liu, 2022) sets the estimated vector as  $\mathcal{P}_G(\mathbf{A}^T\mathbf{y} / m)$ , where  $\mathcal{P}_G$  denotes the projection operator onto the range of  $G$ . Since performing the projection step can be time-consuming even for conventional generative models like GANs, the authors of (Raj et al., 2019) use the composition of the pseudoinverse of the generator and the generator itself to approximate the projection operator. Building on these works, to reconstruct the direction of the signal  $x^{*}$  from the knowledge of the measurement matrix  $\mathbf{A}\in \mathbb{R}^{m\times n}$  and the observed vector  $\pmb {y}\in \mathbb{R}^{m}$  (despite the unknown link  $f$ ), a natural idea is to set the estimated vector  $\hat{x}$  as follows:
+
+$$
+\hat {\boldsymbol {x}} = G \circ G ^ {\dagger} \left(\frac {1}{m} \mathbf {A} ^ {T} \boldsymbol {y}\right). \tag {22}
+$$
+
+However, such a simple idea does not fully exploit the relationship between  $\frac{1}{m}\mathbf{A}^T\mathbf{y}$  and the underlying signal  $\mathbf{x}^*$ , and it leads to unsatisfactory empirical results for diffusion models, as we will see in Section 5. In particular, in the approach corresponding to Eq. (22), the inversion step starts from time  $\epsilon$  (for an  $\epsilon$  close to 0; see Section 2.1), and this implicitly assumes that  $\frac{1}{m}\mathbf{A}^T\mathbf{y}$  approximately follows the target data distribution  $q_0$ . However, based on the following lemmas, we observe that  $\frac{1}{m}\mathbf{A}^T\mathbf{y}$  should instead be regarded as a noisy version of  $\mathbf{x}^* \sim q_0$ , especially when the number of samples  $m$  is relatively small compared to the ambient dimension  $n$ . First, it is easy to obtain the following lemma.
+
+Lemma 1. If  $\epsilon \sim \mathcal{N}(\mathbf{0},\mathbf{I}_n)$ , then we have that the following holds with high probability<sup>5</sup>
+
+$$
+\| \epsilon \| _ {\infty} \leq C \sqrt {\log (2 n)}, \tag {23}
+$$
+
+where  $C$  is a sufficiently large positive constant.
+
+Furthermore, we have the following lemma, which demonstrates that  $\frac{1}{m}\mathbf{A}^T\mathbf{y}$  is a noisy version of  $\mu \mathbf{x}^{*}$  (see Eq. (20) for the definition of  $\mu$ ) and approximately characterizes the noise level. The proofs of Lemmas 1 and 2 are placed in Appendix B.
+
+Lemma 2. Under conditions for the link function  $f$  in Eqs. (20) and (21), we have the following holds w.h.p.:
+
+$$
+\left\| \frac {1}{m} \mathbf {A} ^ {T} \boldsymbol {y} - \mu \boldsymbol {x} ^ {*} \right\| _ {\infty} \leq \frac {C ^ {\prime} \sqrt {\log (2 n)}}{\sqrt {m}}, \tag {24}
+$$
+
+where  $C^\prime$  is a sufficiently large positive constant.
+
+The comparison between Lemmas 1 and 2 inspires us to express  $\frac{1}{m\mu}\mathbf{A}^T\mathbf{y}$  as
+
+$$
+\frac {1}{m \mu} \mathbf {A} ^ {T} \boldsymbol {y} \approx \boldsymbol {x} ^ {*} + \frac {C ^ {\prime}}{C \mu \sqrt {m}} \boldsymbol {\epsilon} \tag {25}
+$$
+
+for  $\epsilon \sim \mathcal{N}(\mathbf{0},\mathbf{I}_n)$ . Additionally, from Eq. (3), we know that  $\pmb{x}_t$  can be written as  $\pmb{x}_t = \alpha_t\pmb{x}_0 + \sigma_t\pmb{\epsilon}_t = \alpha_t(\pmb{x}_0 + (\sigma_t / \alpha_t)\pmb{\epsilon}_t)$  with  $\pmb{\epsilon}_t \sim \mathcal{N}(\mathbf{0},\mathbf{I}_n)$ . Since the nonlinear link function  $f$  is unknown (and consequently,  $\mu$  is also unknown), and  $C, C'$  are undetermined positive constants, we employ a tuning parameter  $C_s$  and find a time  $t^*$  such that
+
+$$
+\frac {\sigma_ {t ^ {*}}}{\alpha_ {t ^ {*}}} = \frac {C _ {s}}{\sqrt {m}}, \tag {26}
+$$
+
+and start the inversion from time  $t^*$  with the input vector being  $\alpha_{t^*} C_s' \mathbf{A}^T \mathbf{y} / m$ , where  $C_s' > 0$  is a tuning parameter. Remark 2. Unlike prior works such as (Chung et al., 2022b; Fabian et al., 2024; Wu et al., 2024) that identify an intermediate time for sampling, we seek an intermediate time mainly for the inversion of DMs. Moreover, this intermediate time is computed using theoretical findings in Lemma 2.
+
+Then, the corresponding estimated vector is
+
+$$
+\hat {\boldsymbol {x}} = G \circ G _ {t ^ {*}} ^ {\dagger} \left(\frac {\alpha_ {t ^ {*}} C _ {s} ^ {\prime}}{m} \mathbf {A} ^ {T} \boldsymbol {y}\right), \tag {27}
+$$
+
+where  $G_{t^*}^\dagger$  denotes the partial inversion operator (from  $t^*$  to  $T$ ; see Eq. (19)). One may also notice that since  $\frac{1}{m}\mathbf{A}^T\mathbf{y}$  can be regarded as a noisy version of  $\mathbf{x}^*$  and the sampling process has a denoising effect, we can simply sample from  $t^*$  to  $\epsilon$  without taking the inversion step. In this case, the corresponding estimated vector is
+
+$$
+\hat {\boldsymbol {x}} = G _ {t ^ {*}} \left(\frac {\alpha_ {t ^ {*}} C _ {s} ^ {\prime}}{m} \mathbf {A} ^ {T} \boldsymbol {y}\right), \tag {28}
+$$
+
+where  $G_{t^*}$  denotes the partial sampling operator (from  $t^*$  to  $\epsilon$ ; see Eq. (15)). Indeed, such a simple and efficient approach can also lead to reasonably good reconstructions, although its performance is inferior to that of the approach corresponding to Eq. (27).
+
+We refer to the approach for Eq. (22) as SIM-DMFIS (SIMs using diffusion models with full inversion and sampling
+
+procedures), the approach for Eq. (28) as SIM-DMS since it only performs a (partial) sampling procedure, and the approach for Eq. (27) as SIM-DMIS. We illustrate the differences among these three approaches in Figure 1. For convenience, we present the SIM-DMIS approach in Algorithm 1.
+
+![](images/2a42134524febdfef4d8d893a65f317d9a6d8ebfbb0885e5eb97871b686f65f4.jpg)  
+(a) SIM- DMS
+
+![](images/eadbf5fcd029e2145ae790e6741a16b6ec1746d4516585f40db0d854d3b2f346.jpg)  
+(b) SIM-DMFIS  
+Figure 1. An illustration of our three approaches. For SIM-DMS, we only perform the sampling from  $t^*$  to  $\epsilon$ . For SIM-DMFIS, we perform the full inversion and sampling procedures. For SIM-DMIS, we first perform the inversion from  $t^*$  to  $T$ , and then perform the sampling from  $T$  to  $\epsilon$ .
+
+![](images/bc6f481a9475f50eb7f0276b5ebdd6444293321b3a8d1757d0313115f1f32524.jpg)  
+(c) SIM-DMIS
+
+# Algorithm 1 The SIM-DMIS approach
+
+Input:  $\mathbf{A} \in \mathbb{R}^{m \times n}$ ,  $\pmb{y} \in \mathbb{R}^m$ ,  $\pmb{x}_{\theta}$ , time steps  $\epsilon = t_N < t_{N-1} < \ldots < t_1 < t_0 = T$ , generator  $G$  corresponding to the sampling process as in Eq. (14), partial inversion operator  $G_t^\dagger$  as in Eq. (19), tuning parameters  $C_s$  and  $C_s'$
+
+1: Calculate  $t^{*}\in [\epsilon ,T]$  as in Eq. (26).
+
+2: Calculate the estimated vector  $\hat{\pmb{x}}$  as in Eq. (27).
+
+Output:  $\hat{x}$
+
+# 4. Theoretical Analysis
+
+In this section, we provide a theoretical analysis of the effectiveness of our proposed SIM-DMIS approach. Before presenting the analysis, we first present some preliminary results. Firstly, the following assumption states that the neural function  $x_{\theta}(x,t)$  is Lipschitz continuous with respect to its first argument. This assumption has been widely adopted in relevant works such as (Lu et al., 2022a; Chen et al., 2023a;c; 2024; 2023d; Bortoli, 2022; Lee et al., 2022; Li et al., 2023).
+
+Assumption 1 (Lipschitz Continuity of the Neural Function). For any  $t \in [0, T]$ , there exists a positive constant  $L_t > 0$  such that the neural function  $\pmb{x}_{\theta}(\pmb{x}, t)$  is  $L_t$ -Lipschitz continuous with respect to its first parameter  $\pmb{x}$ . That is, it holds for all  $\pmb{x}_1, \pmb{x}_2 \in \mathbb{R}^n$  that  $\| \pmb{x}_{\theta}(\pmb{x}_1, t) - \pmb{x}_{\theta}(\pmb{x}_2, t) \|_2 \leq L_t \| \pmb{x}_1 - \pmb{x}_2 \|_2$ .
+
+Based on Assumption 1, it can be easily demonstrated that the generator corresponding to popular sampling methods such as DDIM and DM2M (cf. Sec. 2.1) is Lipschitz continuous. Specifically, we have the following lemma.
+
+Lemma 3. Suppose that the data prediction network  $\pmb{x}_{\theta}$  satisfies Assumption 1. Then, if  $G: \mathbb{R}^n \to \mathbb{R}^n$  is the generator as in Eq. (14), we have that  $G$  is  $L$ -Lipschitz continuous with  $L > 0$  being dependent on  $\{\alpha_{t_i}\}_{i=0}^{N-1}$ ,  $\{\sigma_{t_i}\}_{i=0}^{N-1}$ , and  $\{L_{t_i}\}_{i=0}^{N-1}$ .
+
+Lemma 3 shows that it is reasonable to assume that  $G$  is Lipschitz continuous. The proof of Lemma 3 and the precise characterization of the Lipschitz constant  $L$  for DDIM and DM2M are deferred to Appendix C.
+
+Then, we have the following theorem regarding the effectiveness of the SIM-DMIS approach as presented in Eq. (27). Specifically, for  $t \in [\epsilon, T]$ , it offers an upper bound for the distance between  $G \circ G_t^\dagger(\bar{\boldsymbol{x}}_t)$  and  $\bar{\boldsymbol{x}}_\epsilon$ , where  $\bar{\boldsymbol{x}}_t$  is a sample of  $q_t$  and  $\bar{\boldsymbol{x}}_\epsilon$  is on the same ODE trajectory as  $\bar{\boldsymbol{x}}_t$ . When  $\epsilon$  is sufficiently small and the (scaled) data prediction network  $\boldsymbol{x}_\theta$  provides a good approximation to the ground-truth score  $\nabla \log q_t$  (Chen et al., 2023d; Li et al., 2023),  $\bar{\boldsymbol{x}}_\epsilon$  will be close to the ground-truth data following  $q_0$  that corresponds to  $\bar{\boldsymbol{x}}_t$ . Additionally, it has been demonstrated in (Lu et al., 2022a;b) that under certain regularity conditions, DDIM and DM2M are first- and second-order numerical sampling methods for Eq. (10), respectively. Moreover, by considering the first- and second-order numerical discretization of Eq. (10) from  $t = \epsilon$  to  $t = T$ , we similarly obtain the first- and second-order numerical inversion methods for Eq. (10) respectively (for instance, the inversion method in Eq. (17) is a first-order numerical inversion method).
+
+Therefore, based on Theorem 3, if the term  $\alpha_{t^*}C_s'\mathbf{A}^T\pmb {y} / m$  in Eq. (27) can be approximately expressed as  $\alpha_{t^*}\pmb{x}^* +$ $\sigma_{t^{*}}\epsilon$  with  $\epsilon$  being a standard normal vector,  $\hat{\pmb{x}}\coloneqq G\circ$ $G_{t^*}^\dagger (\alpha_{t^*}C_s'\mathbf{A}^T\pmb {y} / m)$  is close to  $\pmb{x}^*$  under appropriate conditions. The proof of Theorem 3 can be found in Appendix C.
+
+Theorem 3. For  $t \in [\epsilon, T]$ , let  $\bar{\mathbf{x}}_t \in \mathbb{R}^n$  be a sample of  $q_t$ , and let  $\bar{\mathbf{x}}_\epsilon = \frac{\sigma_\epsilon}{\sigma_t} \bar{\mathbf{x}}_t + \sigma_\epsilon \int_{\lambda_t}^{\lambda_\epsilon} e^\lambda \hat{\mathbf{x}}_\theta (\hat{\mathbf{x}}_\lambda, \lambda) \, \mathrm{d}\lambda$ , which is the analytic solution of Eq. (10) with respect to the initial vector  $\bar{\mathbf{x}}_t$ . For  $k_1, k_2 \in \{1, 2, 3\}$ , suppose that  $G^\dagger$  corresponds to a  $k_1$ -th order numerical inversion method for Eq. (10), and  $G_t^\dagger$  denotes the partial inversion operator from  $t$  to  $T$  (see Eq. (19)). Suppose that the generator  $G$  corresponds to a  $k_2$ -th order numerical sampling method for Eq. (10) and  $G$  is  $L$ -Lipschitz continuous. Then, under certain regularity conditions, we have the following:
+
+$$
+\left\| \bar {\boldsymbol {x}} _ {\epsilon} - G \circ G _ {t} ^ {\dagger} (\bar {\boldsymbol {x}} _ {t}) \right\| _ {2} = O \left(\sqrt {n} \left(h _ {\max } ^ {k _ {2}} + L h _ {\max } ^ {k _ {1}}\right)\right), \tag {29}
+$$
+
+where  $h_{\mathrm{max}} = \max_{i\in [N]}(\lambda_{t_i} - \lambda_{t_{i - 1}})$ .
+
+# 5. Experiments
+
+In this section, we conduct a series of experiments to validate the effectiveness of the proposed SIM-DMIS approach (see Algorithm 1). Specifically, we evaluate our method on two datasets: FFHQ  $256 \times 256$  (Karras et al., 2019), ImageNet  $256 \times 256$  (Deng et al., 2009). For FFHQ and Im
+
+ageNet, the ambient dimension is  $n = 3 \times 256 \times 256 = 196608$ .
+
+We perform experiments on nonlinear measurement models with 1-bit and cubic measurements, for which we have  $\pmb{y} = \mathrm{sign}(\mathbf{A}\pmb{x}^{*} + \pmb{e})$  and  $\pmb{y} = (\mathbf{A}\pmb{x}^{*})^{3} + \pmb{e}$  respectively, where  $\pmb{e} \sim \mathcal{N}(\mathbf{0},\sigma^2\mathbf{I}_m)$ . We compare SIM-DMIS with QCS-SGM proposed in (Meng & Kabashima, 2022), which uses DMs to handle quantized compressed sensing problems. We also compare our approach with the posterior sampling methods DPS (Chung et al., 2023b) and DAPS (Zhang et al., 2024a). DPS is a popular baseline for general signal recovery problems, and DAPS has shown excellent performance in nonlinear signal recovery problems. When DPS and DAPS utilize the knowledge of the link function  $f$ , the corresponding methods are denoted as DPS-N and DAPS-N respectively. In addition, we compare with DPS and DAPS for the linear operator without using the knowledge of  $f$ , and the corresponding methods are denoted as DPS-L and DAPS-L respectively.
+
+For the QCS-SGM method, we adhere to the experimental settings specified in (Meng & Kabashima, 2022) to ensure a fair comparison. Additionally, we compare with SIM-DMFIS and SIM-DMS (see Eqs. (22) and (28)), which are two variants of SIM-DMIS. We employ three metrics to assess the performance of these methods: Peak signal-to-noise ratio (PSNR), structural similarity index measure (SSIM), and learned perceptual image patch similarity (LPIPS). The reported quantitative results are averaged over 100 testing images. In the experimental process, we use DDIM for sampling and the DM2M inversion method (refer to Sec. 2.1 for details). The reported NFEs represent the total number of NFEs utilized by each approach. For instance, in the case of SIM-DMIS and SIM-DMFIS, the reported NFE is the sum of the number of steps in the inversion process and the number of steps in the sampling process. We adopt the variance preserving (VP) schedule with  $\beta_{\mathrm{min}} = 0.1$  and  $\beta_{\mathrm{max}} = 20$ . For SIM-DMS, we use 50 NFEs, while for SIM-DMIS and SIM-DMFIS, we use 150 NFEs. $^{10}$  The remaining parameters are set as  $\epsilon = 0.001$ ,  $T = 1$ , and  $\sigma = 0.05$ . For the detailed parameter settings of  $C_s$  and  $C_s'$  in SIM-DMS and SIM-DMIS, please refer to Appendices F and G.
+
+Results for FFHQ  $(256\times 256)$  Since the experiments for FFHQ are time-consuming, we only report the results for  $m = n / 8 = 24576$ . For the QCS-SGM method, we leverage a pre-trained unconditional Score-SDE model with the
+
+For 1-bit measurements where  $f$  is not always differentiable, PyTorch can still enforce automatic differentiation.  
+10The same number of NFEs is used for the inversion process of SIM-DMIS and SIM-DMFIS. As demonstrated in Appendices F and G, this choice has a negligible impact on the experimental results.
+
+variance exploding (VE) schedule<sup>11</sup> for the FFHQ dataset. As for the other methods, we employ the model from DPS, which has been trained on 49,000 FFHQ  $256 \times 256$  images and validated on the first 1,000 images.
+
+It is evident from Table 1 and Figure 2 that SIM-DMIS exhibits superior reconstruction performance with 1-bit measurements. Notably, SIM-DMIS requires only 150 NFEs, in contrast to QCS-SGM, which demands a significantly higher 11,555 NFEs. Additionally, both DPS and DAPS utilize 1,000 NFEs. The results for cubic measurements presented in Table 2 and Figure 3 further demonstrate the effectiveness of our proposed methods. In particular, SIM-DMIS consistently achieves the highest reconstruction quality across all evaluated metrics while requiring only 150 NFEs, demonstrating both the efficiency and robustness of our approach under the cubic measurement setting on the FFHQ dataset.
+
+Note that in 1-bit measurements, even in the noiseless scenario, the link function  $f(x) = \mathrm{sign}(x)$  is non-differentiable at  $x = 0$  (though as mentioned in Footnote 9, PyTorch can still enforce automatic differentiation). The non-differentiability also poses challenges for DPS-N and DAPS-N as they rely on  $f$  in gradient based updates. This can lead to inaccurate gradients and ultimately resulting in subpar performance. $^{12}$  For cubic measurements, since we have observed from the results for 1-bit measurements that SIM-DMFIS, DPS-L, and DAPS-L do not perform well, we do not compare against them. Furthermore, we do not include comparisons with DAPS-N since although the link function for cubic measurements is differentiable, its pronounced non-linearity produces unstable gradient directions that make optimization with DAPS-N unreliable and significantly degrade reconstruction performance.
+
+Table 1. Quantitative results for FFHQ with 1-bit measurements  $(m = n / 8 = 24576)$ . We mark bold the best scores, and underline the second best scores.  
+
+<table><tr><td>Method</td><td>NFE</td><td>PSNR (↑)</td><td>SSIM (↑)</td><td>LPIPS (↓)</td></tr><tr><td>QCS-SGM</td><td>11555</td><td>12.91±2.36</td><td>0.51±0.07</td><td>0.50±0.08</td></tr><tr><td>DPS-N</td><td>1000</td><td>11.14±1.46</td><td>0.37±0.09</td><td>0.69±0.05</td></tr><tr><td>DPS-L</td><td>1000</td><td>8.57±2.05</td><td>0.22±0.08</td><td>0.69±0.09</td></tr><tr><td>DAPS-N</td><td>1000</td><td>16.59±0.54</td><td>0.33±0.05</td><td>0.48±0.05</td></tr><tr><td>DAPS-L</td><td>1000</td><td>5.63±0.71</td><td>0.04±0.03</td><td>0.61±0.03</td></tr><tr><td>SIM- DMS</td><td>50</td><td>17.14±2.41</td><td>0.44±0.07</td><td>0.48±0.05</td></tr><tr><td>SIM-DMFIS</td><td>150</td><td>8.48±0.13</td><td>0.03±0.00</td><td>0.90±0.02</td></tr><tr><td>SIM-DMIS</td><td>150</td><td>19.87±2.77</td><td>0.60±0.09</td><td>0.37±0.05</td></tr></table>
+
+<sup>11</sup>https://huggingface.co/google/ncsnpp-ffhq-256  
+<sup>12</sup>The supplementary material of DAPS suggests using Metropolis Hasting for non-differentiable forward operators, but it is also mentioned to have inferior performance and low efficiency, with results only in the supplementary material.
+
+Table 2. Quantitative results on the FFHQ with cubic measurements ( $m = n/8 = 24576$ ). We mark **bold** the best scores, and underline the second best scores.  
+
+<table><tr><td>Method</td><td>NFE</td><td>PSNR (↑)</td><td>SSIM (↑)</td><td>LPIPS (↓)</td></tr><tr><td>DPS-N</td><td>1000</td><td>15.41±1.38</td><td>0.37±0.08</td><td>0.50±0.07</td></tr><tr><td>SIM-DMS</td><td>50</td><td>14.44±2.13</td><td>0.51±0.10</td><td>0.47±0.08</td></tr><tr><td>SIM-DMIS</td><td>150</td><td>17.56±2.56</td><td>0.57±0.10</td><td>0.40±0.07</td></tr></table>
+
+![](images/39dd09512abd79b6114ad3ade46e756272148592f417b72673d3e3b2a7205f49.jpg)  
+Figure 2. Examples of 1-bit reconstructed images for FFHQ.
+
+![](images/856e64f8569120b8c9091c3669dbc58781dc3a2add696a7ce4ab812b4da985ba.jpg)  
+Figure 3. Examples of cubic reconstructed images for FFHQ.
+
+Results for ImageNet  $(256\times 256)$  For ImageNet, we report the results for  $m = n / 16 = 12288$ . We utilize a pretrained conditional ImageNet  $256\times 256$  model along with its corresponding classifier sourced from ADM (Dhariwal & Nichol, 2021). We adhere to the default recommended configuration settings of these models, substituting the unconditional ImageNet  $256\times 256$  model initially provided by DPS. The tests are conducted on 100 images selected from the Validation1K Set. Given that QCS-SGM lacks pre-trained models for ImageNet, we do not compare with it.
+
+Table 3 and Figure 4 demonstrate that SIM-DMIS achieves superior reconstruction with 1-bit measurements compared to DPS and DAPS variants, while requiring only 150 NFEs. The results with cubic measurements, summarized in Table 4 and Figure 5, further validate the advantages of our methods. Notably, SIM-DMIS achieves the best performance across all metrics while maintaining a low computational cost of only 150 NFEs. These results underscore the effectiveness and efficiency of our approach in handling cubic measurement settings on the ImageNet dataset.
+
+Table 3. Quantitative results for ImageNet with 1-bit measurements ( $m = n/16 = 12288$ ). We mark **bold** the best scores, and **underline** the second best scores.  
+
+<table><tr><td>Method</td><td>NFE</td><td>PSNR (↑)</td><td>SSIM (↑)</td><td>LPIPS (↓)</td></tr><tr><td>DPS-N</td><td>1000</td><td>11.57±2.63</td><td>0.24±0.11</td><td>0.64±0.04</td></tr><tr><td>DPS-L</td><td>1000</td><td>12.67±3.43</td><td>0.11±0.16</td><td>0.81±0.12</td></tr><tr><td>DAPS-N</td><td>1000</td><td>14.62±1.20</td><td>0.12±0.01</td><td>0.61±0.04</td></tr><tr><td>DAPS-L</td><td>1000</td><td>6.20±1.66</td><td>0.08±0.14</td><td>0.64±0.02</td></tr><tr><td>SIM-DMS</td><td>50</td><td>16.93±3.16</td><td>0.32±0.09</td><td>0.54±0.09</td></tr><tr><td>SIM-DMFIS</td><td>150</td><td>11.89±2.24</td><td>0.08±0.02</td><td>0.76±0.06</td></tr><tr><td>SIM-DMIS</td><td>150</td><td>18.06±3.16</td><td>0.45±0.13</td><td>0.46±0.08</td></tr></table>
+
+Table 4. Quantitative results on the ImageNet with cubic measurements ( $m = n/16 = 12288$ ). We mark **bold** the best scores, and **underline** the second best scores.  
+
+<table><tr><td>Method</td><td>NFE</td><td>PSNR (↑)</td><td>SSIM (↑)</td><td>LPIPS (↓)</td></tr><tr><td>DPS-N</td><td>1000</td><td>11.47±2.97</td><td>0.22±0.08</td><td>0.62±0.06</td></tr><tr><td>SIM-DMS</td><td>50</td><td>14.99±2.82</td><td>0.33±0.06</td><td>0.59±0.08</td></tr><tr><td>SIM-DMIS</td><td>150</td><td>16.28±3.38</td><td>0.40±0.15</td><td>0.55±0.10</td></tr></table>
+
+![](images/038a0d162186e703ede681055302880d493906ea91c6152e2e94dcb5534afc40.jpg)  
+Figure 4. Examples of 1-bit reconstructed images for ImageNet.
+
+![](images/da8f1d5f175d6ace58dc53825e217c4bb703da30bfc09b175197e5bc8bf6e7b0.jpg)  
+Figure 5. Examples of cubic reconstructed images for ImageNet.
+
+# 6. Conclusion
+
+In this paper, we present approaches for learning SIMs by making use of the sampling and inversion methods for pre-trained unconditional DMs. Theoretical analysis and numerical results are provided to illustrate the effectiveness of the proposed approaches.
+
+# Impact Statement
+
+This paper presents work aimed at advancing the field of Machine Learning, particularly in the area of signal recovery using diffusion models (DMs). The proposed method applies DMs to enhance the recovery of signals from Single Index Models (SIMs), offering more flexible and robust solutions for complex signal reconstruction tasks. It could have broad applications in fields such as imaging, signal processing, and data reconstruction, with potential to improve the efficiency and accuracy of various real-world tasks. While no direct ethical concerns are associated with the methodology, the potential societal impacts of this work include improvements in computational efficiency and the enhancement of technologies reliant on image reconstruction and signal recovery.
+
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+
+# A. Extended Related Work
+
+In this subsection, we summarize some relevant works, which can roughly be divided into (i) nonlinear signal recovery with conventional generative models, and (ii) signal recovery with diffusion models.
+
+Nonlinear signal recovery with conventional generative models: The seminal work (Bora et al., 2017) studies linear compressed sensing with conventional generative priors (such as GANs and VAEs) and demonstrates via numerical experiments on image datasets that using a pre-trained generative prior can significantly reduce the number of required measurements (compared to that of using the sparse prior) for accurate signal recovery. This has led to a significant volume of follow-up works in (Dhar et al., 2018; Hand et al., 2018; Shah & Hegde, 2018; Jagatap & Hegde, 2019; Asim et al., 2020; Ongie et al., 2020; Daras et al., 2021; Liu & Scarlett, 2020b; Jalal et al., 2021b;a), which explore various aspects of high-dimensional signal recovery with generative priors. A literature review in this area can be found in (Scarlett et al., 2022).
+
+In particular, under conventional generative priors, there have been several studies on 1-bit compressed sensing (Qiu et al., 2020; Liu et al., 2020; 2021a) and phase retrieval (Hand et al., 2018; Hyder et al., 2019; Jagatap & Hegde, 2019; Aubin et al., 2020; Shamshad & Ahmed, 2020; Liu et al., 2021b; Killedar & Seelamantula, 2022; Liu et al., 2022b), among others. Moreover, SIMs that account for unknown nonlinearity and conventional generative priors have been explored in several studies, including (Wei et al., 2019; Liu & Scarlett, 2020a; Liu & Liu, 2022; Chen et al., 2023b). More specifically, the works (Liu & Scarlett, 2020a; Liu & Liu, 2022; Chen et al., 2023b) provide recovery guarantees for the generalized Lasso approach and practical algorithms under the assumption of Gaussian sensing vectors. In (Wei et al., 2019), under the extra assumption that the link function is differentiable, SIMs with link functions of first- and second-orders are studied through Stein's identity, allowing for general non-Gaussian sensing vectors.
+
+Signal recovery with diffusion models: Due to the ability to model complex distributions, DMs have demonstrated remarkable performance in a variety of applications. The seminal work on DMs (Sohl-Dickstein et al., 2015) introduces a framework for the diffusion process, which has been further developed in multiple forms, such as denoising diffusion probabilistic models (DDPMs) (Ho et al., 2020) and score-based generative models (SGMs) (Song et al., 2021b). These models have been widely applied to signal recovery problems (Daras et al., 2024), which can be broadly classified into two main paradigms.
+
+One paradigm is task-specific, in which DMs are trained for particular tasks. For example, the model in (Saharia et al., 2022c) is customized for super-resolution, while the model in (Ren et al., 2023) is developed for deblurring. The other paradigm makes use of pre-trained DMs that are not limited to any specific signal recovery problem. Works within this paradigm can be further classified according to how they handle the measurements. Some approaches use DMs as priors to direct the reconstruction process by estimating the score function, as shown in (Feng et al., 2023; Daras et al., 2022). Other approaches, such as DDRM (Kawar et al., 2022), adapt the DDPM framework specifically for linear signal recovery problems. By incorporating singular value decomposition to limit the solution space, DDRM can effectively handle linear measurements. However, its performance degrades when dealing with sparse or limited measurements, rendering it less effective in challenging situations.
+
+To solve general noisy linear signal recovery problems, some recent works (Chung et al., 2022a; 2023b; Song et al., 2023; Fei et al., 2023; Fabian et al., 2023; Zhang et al., 2023) aim to estimate the posterior distribution using the unconditional diffusion model based on Bayes' rule. In particular, MCG (Chung et al., 2022a) approaches data consistency from the perspective of the data manifold. It proposes a manifold-constrained gradient to ensure that corrections stay on the data manifold DPS (Chung et al., 2023b) abandons the projection step in the reverse process because it may cause the sampling path to deviate from the data manifold. Instead, it approximates the gradient of the posterior through a specially hand-designed strategy, where the measurements can be regarded as a signal conditioning the sampling process. IIGDM (Song et al., 2023) further develops this type of strategy into a unified expression that encompasses various signal recovery problems by using the Moore-Penrose pseudoinverse. IIGDM guides the diffusion process by matching the one-step denoising solution and the ground-truth measurements, after transforming both via a generalized pseudoinverse of the measurement model. DAPS (Zhang et al., 2024a) modifies the traditional diffusion process by decoupling consecutive steps in the sampling trajectory. This allows for greater variation between steps, improving the exploration of the solution space and leading to more accurate and stable reconstructions.
+
+# B. Proofs of Lemmas 1 and 2
+
+Before presenting the proofs, we present the following simple tail bound for a Gaussian random variable.
+
+Lemma 4. (Gaussian tail bound (Wainwright, 2019, Example 2.1)) Suppose that  $X \sim \mathcal{N}(\alpha, \sigma^2)$  is a Gaussian random variable with mean  $\alpha$  and variance  $\sigma^2$ . Then, for any  $u > 0$ ,  $\mathbb{P}(|X - \alpha| \geq u) \leq 2e^{-\frac{u^2}{2\sigma^2}}$ .
+
+# B.1. Proof of Lemma 1
+
+For  $u > 0$  and any  $j \in [n]$ , from Lemma 4, from Lemma 4, we have with probability at least  $1 - 2e^{-\frac{u^2}{2}}$  that  $|\epsilon_j| \leq u$ . Taking a union bound over all  $j \in [n]$ , we obtain with probability at least  $1 - 2ne^{-\frac{u^2}{2}}$  that  $\| \epsilon \|_{\infty} \leq u$ . Then, by setting  $u = C\sqrt{\log(2n)}$ , we obtain the desired result.
+
+# B.2. Proof of Lemma 2
+
+Let  $g_{i} = \pmb{a}_{i}^{T}\pmb{x}^{*}$ . Since  $\pmb{x}^{*} \in S^{n - 1}$ , we know that  $g_{i}$  are independent standard normal random variables. Additionally, note that
+
+$$
+\begin{array}{l} \frac {1}{m} \mathbf {A} ^ {T} \boldsymbol {y} - \mu \boldsymbol {x} ^ {*} = \frac {1}{m} \sum_ {i = 1} ^ {m} y _ {i} \boldsymbol {a} _ {i} - \mu \boldsymbol {x} ^ {*} (30) \\ = \frac {1}{m} \sum_ {i = 1} ^ {m} y _ {i} \left(\boldsymbol {a} _ {i} - \left(\boldsymbol {a} _ {i} ^ {T} \boldsymbol {x} ^ {*}\right) \boldsymbol {x} ^ {*}\right) + \frac {1}{m} \sum_ {i = 1} ^ {m} y _ {i} \left(\boldsymbol {a} _ {i} ^ {T} \boldsymbol {x} ^ {*}\right) \boldsymbol {x} ^ {*} - \mu \boldsymbol {x} ^ {*}. (31) \\ \end{array}
+$$
+
+For any  $j \in [n]$ , since  $\mathbb{E}[a_{ij}g_i] = x_j^*$ ,  $a_{ij}$  can be written as
+
+$$
+a _ {i j} = x _ {j} ^ {*} g _ {i} + \sqrt {1 - \left(x _ {j} ^ {*}\right) ^ {2}} r _ {i j}, \tag {32}
+$$
+
+where  $r_{ij}$  is a standard normal random variable that is independent of  $g_i$ . Then, the  $j$ -th entry of the first term in the right-hand side of Eq. (31) can be written as
+
+$$
+\begin{array}{l} \left[ \frac {1}{m} \sum_ {i = 1} ^ {m} y _ {i} \left(\boldsymbol {a} _ {i} - \left(\boldsymbol {a} _ {i} ^ {T} \boldsymbol {x} ^ {*}\right) \boldsymbol {x} ^ {*}\right) \right] _ {j} = \frac {1}{m} \sum_ {i = 1} ^ {m} y _ {i} \left(a _ {i j} - g _ {i} x _ {j} ^ {*}\right) (33) \\ = \sqrt {1 - \left(x _ {j} ^ {*}\right) ^ {2}} \cdot \frac {1}{m} \sum_ {i = 1} ^ {m} y _ {i} r _ {i j}. (34) \\ \end{array}
+$$
+
+Note that  $y_{i} = f(\pmb{a}_{i}^{T}\pmb{x}^{*}) = f(g_{i})$ . Thus  $r_{ij}$  is also independent of  $y_{i}$ . Let  $M_2 \coloneqq \mathbb{E}_{g\sim \mathcal{N}(0,1)}[f(g)^2]$  and  $\mathcal{E}_1$  be the event that  $\frac{1}{m}\sum_{i=1}^{m}y_{i}^{2} \leq 2M_{2}$ . From (Liu & Liu, 2022, Lemma 3), we have
+
+$$
+\mathbb {P} \left(\mathcal {E} _ {1} ^ {c}\right) \leq \frac {C _ {f}}{m}, \tag {35}
+$$
+
+where  $C_f$  is a positive constant depending on  $f$ . Moreover, conditioned on the event  $\mathcal{E}_1$ , the term  $\sum_{i=1}^{m} y_i r_{ij}$  in Eq. (34) is zero-mean Gaussian with the variance being  $\sum_{i=1}^{m} y_i^2$ . Then, from Lemma 4, we obtain
+
+$$
+\mathbb {P} \left(\left| \sum_ {i = 1} ^ {m} y _ {i} r _ {i j} \right| > u\right) \leq 2 \exp \left(- \frac {u ^ {2}}{2 \sum_ {i = 1} ^ {m} y _ {i} ^ {2}}\right) \leq 2 \exp \left(- \frac {u ^ {2}}{4 m M _ {2}}\right). \tag {36}
+$$
+
+Taking a union bound over all  $j \in [n]$ , we obtain with probability at least  $1 - 2n\exp \left(-\frac{u^2}{4mM_2}\right)$  that
+
+$$
+\left\| \frac {1}{m} \sum_ {i = 1} ^ {m} y _ {i} \left(\boldsymbol {a} _ {i} - \left(\boldsymbol {a} _ {i} ^ {T} \boldsymbol {x} ^ {*}\right) \boldsymbol {x} ^ {*}\right) \right\| _ {\infty} = \left\| \sqrt {1 - \left(x _ {j} ^ {*}\right) ^ {2}} \cdot \frac {1}{m} \sum_ {i = 1} ^ {m} y _ {i} r _ {i j} \right\| _ {\infty} \leq \frac {u}{m}. \tag {37}
+$$
+
+Furthermore, the  $j$ -th entry of the second term in the right-hand side of Eq. (31) can be written as
+
+$$
+\frac {1}{m} \sum_ {i = 1} ^ {m} y _ {i} \left(\boldsymbol {a} _ {i} ^ {T} \boldsymbol {x} ^ {*}\right) \boldsymbol {x} ^ {*} - \mu \boldsymbol {x} ^ {*} = \left(\frac {1}{m} \sum_ {i = 1} ^ {m} y _ {i} \left(\boldsymbol {a} _ {i} ^ {T} \boldsymbol {x} ^ {*}\right) - \mu\right) \boldsymbol {x} ^ {*}. \tag {38}
+$$
+
+For any  $\tau > 0$ , let  $\mathcal{E}_2$  be the event that  $\left|\frac{1}{m}\sum_{i=1}^{m}y_i(\pmb{a}_i^T\pmb{x}^*) - \mu\right| < \tau$ . From (Liu & Liu, 2022, Lemma 3), we have
+
+$$
+\mathbb {P} \left(\mathcal {E} _ {2} ^ {c}\right) \leq \frac {C _ {f} ^ {\prime}}{m \tau^ {2}}, \tag {39}
+$$
+
+where  $C_f^\prime$  is a positive constant depending on  $f$ . Additionally, conditioned on  $\mathcal{E}_2$ , we have
+
+$$
+\left\| \left(\frac {1}{m} \sum_ {i = 1} ^ {m} y _ {i} \left(\boldsymbol {a} _ {i} ^ {T} \boldsymbol {x} ^ {*}\right) - \mu\right) \boldsymbol {x} ^ {*} \right\| _ {\infty} = \left| \frac {1}{m} \sum_ {i = 1} ^ {m} y _ {i} \left(\boldsymbol {a} _ {i} ^ {T} \boldsymbol {x} ^ {*}\right) - \mu \right| \cdot \| \boldsymbol {x} ^ {*} \| _ {\infty} \leq \tau . \tag {40}
+$$
+
+Combining Eqs. (31), (35), (37), (39), and (40), we observe that by setting  $u = C_1 \sqrt{\log(2n)} \cdot \sqrt{m}$  and  $\tau = C_2 \sqrt{\log(2n)} / \sqrt{m}$  for some sufficiently large positive constants  $C_1$  and  $C_2$  and letting  $C' = C_1 + C_2$ , we obtain the desired result.
+
+# C. Proof of Theorem 3
+
+First, we provide a proof for the auxiliary result in Lemma 3 for popular sampling methods such as DDIM and DM2M (cf. Sec. 2.1). Specifically, we have the following detailed version of Lemma 3.
+
+Lemma 5. Suppose that the data prediction network  $\pmb{x}_{\pmb{\theta}}$  satisfies Assumption 1. Then, if  $G: \mathbb{R}^n \to \mathbb{R}^n$  is the generator corresponding to the entire sampling process of DDIM (see Eq. (11)), we have that  $G$  is L-Lipschitz continuous with
+
+$$
+L = \prod_ {i = 1} ^ {N} \left(\frac {\sigma_ {t _ {i}}}{\sigma_ {t _ {i - 1}}} + \sigma_ {t _ {i}} \left(\frac {\alpha_ {t _ {i}}}{\sigma_ {t _ {i}}} - \frac {\alpha_ {t _ {i - 1}}}{\sigma_ {t _ {i - 1}}}\right) \cdot L _ {t _ {i - 1}}\right). \tag {41}
+$$
+
+Additionally, suppose that  $G: \mathbb{R}^n \to \mathbb{R}^n$  is the generator corresponding to the entire sampling process of DM2M (see Eq. (12)), and suppose that for  $i = 0, 1, 2, \ldots, N$ ,  $\tilde{\boldsymbol{x}}_{t_i}$  is  $\tilde{L}_i$ -Lipschitz continuous with respect to  $\tilde{\boldsymbol{x}}_{t_0}$ . Then, we have that  $\tilde{L}_0$  can be set to 1 and  $\tilde{L}_1$  can be set to be  $\frac{\sigma_{t_1}}{\sigma_{t_0}} + \sigma_{t_1}\left(\frac{\alpha_{t_1}}{\sigma_{t_1}} - \frac{\alpha_{t_0}}{\sigma_{t_0}}\right) \cdot L_{t_0}$ , and for  $i \geq 2$ ,  $\tilde{L}_i$  can be calculated via the following recursive formula:
+
+$$
+\tilde {L} _ {i} = \left(\frac {\sigma_ {t _ {i}}}{\sigma_ {t _ {i - 1}}} + \alpha_ {t _ {i}} (1 - e ^ {- h _ {i}}) \cdot \left(1 + \frac {1}{2 r _ {i}}\right) L _ {t _ {i - 1}}\right) \tilde {L} _ {i - 1} + \alpha_ {t _ {i}} (1 - e ^ {- h _ {i}}) \cdot \frac {1}{2 r _ {i}} L _ {t _ {i - 2}} \tilde {L} _ {i - 2}. \tag {42}
+$$
+
+In particular, we have that  $G$  is  $L$ -Lipschitz continuous with  $L = \tilde{L}_N$ .
+
+Proof. For DDIM, by the triangle inequality, we have the following inequality for any  $i \in [N]$  and any  $\pmb{x}_1, \pmb{x}_2 \in \mathbb{R}^n$ :
+
+$$
+\left\| \kappa_ {i} \left(\boldsymbol {x} _ {1}\right) - \kappa_ {i} \left(\boldsymbol {x} _ {2}\right) \right\| _ {2} \leq \left(\frac {\sigma_ {t _ {i}}}{\sigma_ {t _ {i - 1}}} + \sigma_ {t _ {i}} \left(\frac {\alpha_ {t _ {i}}}{\sigma_ {t _ {i}}} - \frac {\alpha_ {t _ {i - 1}}}{\sigma_ {t _ {i - 1}}}\right) \cdot L _ {t _ {i - 1}}\right) \cdot \left\| \boldsymbol {x} _ {1} - \boldsymbol {x} _ {2} \right\| _ {2}, \tag {43}
+$$
+
+where  $\kappa_{i}$  is defined in Eq. (13). Additionally, it is easy to show that if function  $f$  is  $L_{f}$ -Lipschitz and  $g$  is  $L_{g}$ -Lipschitz, then their composition  $f\circ g$  is  $(L_{f}L_{g})$ -Lipschitz (Bora et al., 2017). Then, we obtain that the generator  $G = \kappa_{N}\circ \dots \circ \kappa_{2}\circ \kappa_{1}$  (see Eq. (14)) is  $L$ -Lipschitz continuous, with  $L$  given by Eq. (41). Note that for the first sampling step of DM2M, we need to use the first-order numerical scheme—DDIM. Then, similarly to the case of DDIM, based on Eq. (12), we can obtain the desired recursive formula for the Lipschitz constants.
+
+Then, we list the regularity conditions on  $\pmb{x}_{\theta}$  required for Theorem 3 as follows. These conditions are similar to those in (Lu et al., 2022a, Appendix B.1).
+
+The total derivatives  $\frac{\mathrm{d}^j\hat{x}_\theta(\hat{x}_\lambda,\lambda)}{\mathrm{d}\lambda^j}$  (as a function of  $\lambda$ ) exist and are continuous for  $j = 0,1,2,3$  
+- The neural function  $x_{\theta}(x, t)$  is Lipschitz continuous with w.r.t. its first parameter  $x$  (i.e., Assumption 1).  
+$h_{\mathrm{max}} = \max_{i\in [N]}(\lambda_{t_i} - \lambda_{t_{i - 1}}) = O(1 / N).$
+
+Based on the above auxiliary results, we are now ready to present the proof of Theorem 3.
+
+# C.1. Proof of Theorem 3
+
+Let  $\bar{G}_t^\dagger$  be the analytic inversion operator (corresponding to Eq. (10)) from time  $t$  to  $T$ . That is, for any  $\pmb{x} \in \mathbb{R}^n$ , we have
+
+$$
+\bar {G} _ {t} ^ {\dagger} (\boldsymbol {x}) = \frac {\sigma_ {T}}{\sigma_ {t}} \boldsymbol {x} + \sigma_ {T} \int_ {\lambda_ {t}} ^ {\lambda_ {T}} e ^ {\lambda} \hat {\boldsymbol {x}} _ {\boldsymbol {\theta}} (\hat {\boldsymbol {x}} _ {\lambda}, \lambda) \mathrm {d} \lambda . \tag {44}
+$$
+
+Additionally, let  $\bar{G}$  be the analytic sampling operator (corresponding to Eq. (10)) from time  $T$  to  $\epsilon$ . That is, for any  $\pmb{x} \in \mathbb{R}^n$ , we have
+
+$$
+\bar {G} (\boldsymbol {x}) = \frac {\sigma_ {\epsilon}}{\sigma_ {T}} \boldsymbol {x} + \sigma_ {\epsilon} \int_ {\lambda_ {T}} ^ {\lambda_ {\epsilon}} e ^ {\lambda} \hat {\boldsymbol {x}} _ {\boldsymbol {\theta}} \left(\hat {\boldsymbol {x}} _ {\lambda}, \lambda\right) \mathrm {d} \lambda . \tag {45}
+$$
+
+Let  $\bar{\pmb{x}}_T = \bar{G}_t^\dagger (\bar{\pmb{x}}_t) = \frac{\sigma_T}{\sigma_t}\bar{\pmb{x}}_t + \sigma_T\int_{\lambda_t}^{\lambda_T}e^{\lambda}\hat{\pmb{x}}_\pmb {\theta}(\hat{\pmb{x}}_\lambda ,\lambda)\mathrm{d}\lambda$  We have
+
+$$
+\begin{array}{l} \bar {G} \circ \bar {G} _ {t} ^ {\dagger} (\bar {\boldsymbol {x}} _ {t}) = \bar {G} (\bar {\boldsymbol {x}} _ {T}) (46) \\ = \frac {\sigma_ {\epsilon}}{\sigma_ {\epsilon}} \bar {\boldsymbol {x}} _ {T} + \sigma_ {\epsilon} \int_ {\lambda_ {T}} ^ {\lambda_ {\epsilon}} e ^ {\lambda} \hat {\boldsymbol {x}} _ {\boldsymbol {\theta}} (\hat {\boldsymbol {x}} _ {\lambda}, \lambda) \mathrm {d} \lambda (47) \\ = \frac {\sigma_ {\epsilon}}{\sigma_ {T}} \left(\frac {\sigma_ {T}}{\sigma_ {t}} \bar {\boldsymbol {x}} _ {t} + \sigma_ {T} \int_ {\lambda_ {t}} ^ {\lambda_ {T}} e ^ {\lambda} \hat {\boldsymbol {x}} _ {\boldsymbol {\theta}} (\hat {\boldsymbol {x}} _ {\lambda}, \lambda) \mathrm {d} \lambda\right) + \sigma_ {\epsilon} \int_ {\lambda_ {T}} ^ {\lambda_ {\epsilon}} e ^ {\lambda} \hat {\boldsymbol {x}} _ {\boldsymbol {\theta}} (\hat {\boldsymbol {x}} _ {\lambda}, \lambda) \mathrm {d} \lambda (48) \\ = \frac {\sigma_ {\epsilon}}{\sigma_ {t}} \bar {\boldsymbol {x}} _ {t} + \sigma_ {\epsilon} \int_ {\lambda_ {t}} ^ {\lambda_ {T}} e ^ {\lambda} \hat {\boldsymbol {x}} _ {\boldsymbol {\theta}} (\hat {\boldsymbol {x}} _ {\lambda}, \lambda) d \lambda + \sigma_ {\epsilon} \int_ {\lambda_ {T}} ^ {\lambda_ {\epsilon}} e ^ {\lambda} \hat {\boldsymbol {x}} _ {\boldsymbol {\theta}} (\hat {\boldsymbol {x}} _ {\lambda}, \lambda) d \lambda (49) \\ = \frac {\sigma_ {\epsilon}}{\sigma_ {t}} \bar {\boldsymbol {x}} _ {t} + \sigma_ {\epsilon} \int_ {\lambda_ {t}} ^ {\lambda_ {\epsilon}} e ^ {\lambda} \hat {\boldsymbol {x}} _ {\boldsymbol {\theta}} (\hat {\boldsymbol {x}} _ {\lambda}, \lambda) d \lambda (50) \\ = \bar {x} _ {\epsilon}. (51) \\ \end{array}
+$$
+
+Therefore, we obtain
+
+$$
+\begin{array}{l} \left\| G \circ G _ {t} ^ {\dagger} \left(\bar {x} _ {t}\right) - \bar {x} _ {\epsilon} \right\| _ {2} = \left\| G \circ G _ {t} ^ {\dagger} \left(\bar {x} _ {t}\right) - \bar {G} \circ \bar {G} _ {t} ^ {\dagger} \left(\bar {x} _ {t}\right) \right\| _ {2} (52) \\ \leq \| G \circ G _ {t} ^ {\dagger} (\bar {\boldsymbol {x}} _ {t}) - G \circ \bar {G} _ {t} ^ {\dagger} (\bar {\boldsymbol {x}} _ {t}) \| _ {2} + \| G \circ \bar {G} _ {t} ^ {\dagger} (\bar {\boldsymbol {x}} _ {t}) - \bar {G} \circ \bar {G} _ {t} ^ {\dagger} (\bar {\boldsymbol {x}} _ {t}) \| _ {2} (53) \\ \leq L \left\| G _ {t} ^ {\dagger} \left(\bar {\boldsymbol {x}} _ {t}\right) - \bar {G} _ {t} ^ {\dagger} \left(\bar {\boldsymbol {x}} _ {t}\right) \right\| _ {2} + \left\| G \left(\bar {\boldsymbol {x}} _ {T}\right) - \bar {G} \left(\bar {\boldsymbol {x}} _ {T}\right) \right\| _ {2}, (54) \\ \end{array}
+$$
+
+where the last inequality follows from the  $L$ -Lipschitz continuity of  $G$  and  $\bar{\boldsymbol{x}}_T = \bar{G}_t^\dagger (\bar{\boldsymbol{x}}_t)$ . Under the regularity conditions listed above, similarly to (Lu et al., 2022a, Theorem 3.2), we have
+
+$$
+\left\| G _ {t} ^ {\dagger} \left(\bar {\boldsymbol {x}} _ {t}\right) - \bar {G} _ {t} ^ {\dagger} \left(\bar {\boldsymbol {x}} _ {t}\right) \right\| _ {2} = O \left(\sqrt {n} h _ {\max } ^ {k _ {1}}\right) \tag {55}
+$$
+
+and
+
+$$
+\left\| G \left(\bar {x} _ {T}\right) - \bar {G} \left(\bar {x} _ {T}\right) \right\| _ {2} = O \left(\sqrt {n} h _ {\max } ^ {k _ {2}}\right). \tag {56}
+$$
+
+Combining Eqs. (54), (55), and (56), we obtain the desired result.
+
+# D. Experimental Results for CIFAR-10 with 1-bit and cubic Measurements
+
+In this section, we present the experimental results obtained from the CIFAR-10 dataset (Krizhevsky & Hinton, 2009) using 1-bit and cubic measurements. For the CIFAR-10 dataset, the ambient dimension is  $n = 32 \times 32 \times 3 = 3072$ , and the pixel values are normalized to the range [0, 1]. Prior experimental findings have shown that the SIM-DMIS method outperforms SIM-DMS and SIM-DMFIS in terms of reconstruction performance. Therefore, in this study, we limit our comparison to our SIM-DMIS approach against QCS-SGM, DPS-N, and DPS-L. For 1-bit measurements, since QCS-SGM cannot handle cubic measurements, we compare against OneShot proposed in (Liu & Liu, 2022), which uses GANs to solve nonlinear signal recovery problems. Additionally, since we have observed from the results for 1-bit measurements that SIM-DMFIS does not perform well, we do not compare against it. The experimental results demonstrate the superior performance of our proposed method SIM-DMIS across different datasets and measurement settings.
+
+Results on CIFAR-10  $(32\times 32)$  with 1-Bit Measurements For 1-bit measurements, we report results under two measurement settings:  $m = 500$  and  $m = 1000$ , corresponding to approximately  $16\%$  and  $32\%$  of the ambient dimension  $n$ . We utilize a pre-trained unconditional DDPM model with the variance preserving (VP) schedule for CIFAR-10. Prior experimental findings have shown that the SIM-DMIS method outperforms SIM-DMS and SIM-DMFIS in terms of reconstruction performance. Therefore, in this study, we limit our comparison to our SIM-DMIS approach against QCS-SGM, DPS-N, and DPS-L. As shown in Table 5, SIM-DMIS consistently outperforms existing approaches across both  $m = 500$  and  $m = 1000$ . Qualitative results in Figure 6 further confirm its strong reconstruction capability.
+
+Table 5. Quantitative results for 1-bit with  $m = {500}$  and  $m = {1000}$  measurements. We mark bold the best scores, and underline the second best scores.  
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">NFE</td><td colspan="3">m = 500</td><td colspan="3">m = 1000</td></tr><tr><td>PSNR (↑)</td><td>SSIM (↑)</td><td>LPIPS (↓)</td><td>PSNR (↑)</td><td>SSIM (↑)</td><td>LPIPS (↓)</td></tr><tr><td>QCS-SGM</td><td>1160</td><td>8.40±3.75</td><td>0.25±0.19</td><td>0.55±0.20</td><td>11.76±4.59</td><td>0.41±0.22</td><td>0.41±0.18</td></tr><tr><td>DPS-N</td><td>1000</td><td>9.43±1.89</td><td>0.04±0.06</td><td>0.57±0.05</td><td>9.70±2.19</td><td>0.04±0.06</td><td>0.65±0.08</td></tr><tr><td>DPS-L</td><td>1000</td><td>11.53±1.59</td><td>0.08±0.02</td><td>0.61±0.03</td><td>12.24±2.71</td><td>0.10±0.05</td><td>0.69±0.07</td></tr><tr><td>SIM-DMIS</td><td>150</td><td>16.24±1.65</td><td>0.55±0.10</td><td>0.40±0.10</td><td>18.61±1.65</td><td>0.65±0.08</td><td>0.31±0.07</td></tr></table>
+
+![](images/fedec3cf3f112032d91b19b9d5f481ef5338b5c51afa392ab6b8dd2a4f8ab8db.jpg)  
+Figure 6. Examples of reconstructed images for CIFAR-10 with  $m = 1000$  1-bit measurements.
+
+Results on CIFAR-10  $(32\times 32)$  with Cubic Measurements The quantitative results are shown in Table 6, and examples of reconstructed images are presented in Figure 7. SIM-DMIS consistently outperforms OneShot across all measurement
+
+settings. Specifically, with  $m = 500$  measurements, SIM-DMIS achieves approximately  $15\%$  higher PSNR and  $2\%$  better SSIM compared to OneShot. With  $m = 1000$  measurements, SIM-DMIS achieves approximately  $25\%$  higher PSNR and  $12\%$  better SSIM compared to OneShot. Furthermore, SIM-DMIS consistently yields a lower LPIPS value, indicating better perceptual similarity to the original images, compared to OneShot.
+
+Table 6. Quantitative results for CIFAR-10 with cubic measurements. We mark bold the best scores, and underline the second best scores.  
+
+<table><tr><td rowspan="2">Method</td><td colspan="3">m = 500</td><td colspan="3">m = 1000</td></tr><tr><td>PSNR (↑)</td><td>SSIM (↑)</td><td>LPIPS (↓)</td><td>PSNR (↑)</td><td>SSIM (↑)</td><td>LPIPS (↓)</td></tr><tr><td>OneShot</td><td>13.68±2.09</td><td>0.33±0.09</td><td>0.50±0.04</td><td>14.61±2.09</td><td>0.39±0.09</td><td>0.48±0.04</td></tr><tr><td>SIM-DMIS</td><td>15.80±2.02</td><td>0.45±0.10</td><td>0.43±0.08</td><td>16.97±2.21</td><td>0.55±0.12</td><td>0.38±0.09</td></tr></table>
+
+![](images/5c858b4ecd1d080aa10e6540d2c18ed2b10de424dc468dfb504ece5f91d27359.jpg)  
+Figure 7. Examples of reconstructed images for CIFAR-10 with  $m = 1000$  cubic measurements.
+
+# E. Additional Examples of Reconstructed Images for FFHQ and ImageNet with 1-bit Measurements
+
+In this section, we present some additional examples of reconstructed images for the FFHQ  $256 \times 256$  dataset in Figure 8, and also present some additional examples of reconstructed images for the ImageNet  $256 \times 256$  dataset in Figure 9.
+
+![](images/66646f17a9892f12649ff63a1a6d1504297f8b5ecc163b732926980933bf400e.jpg)
+
+![](images/bc51431fd762d1d2ae04142723db80420f5ebac5706062db626a43218ec6c4d5.jpg)
+
+![](images/592431885635cd9692bf1c0e885c12090956e7037d8be802ac70ced39899a853.jpg)  
+Figure 8. Examples of reconstructed images for FFHQ with  $m = n / 8 = 24576$  1-bit measurements. Top: Origin, Middle: SIM-DMS, Bottom: SIM-DMIS.
+
+![](images/d3c3c784826947dcb5e99ce18107a609cdffe2dec809f2a5d72fc4e62d7ced14.jpg)  
+Figure 9. Examples of reconstructed images for ImageNet with  $m = n / 16 = 12288$  1-bit measurements.
+
+# F. Ablation Study for SIM-DMS
+
+In this section, we present a detailed analysis of the ablation experiments conducted on SIM-DMs using the CIFAR-10 dataset with  $m = 500$  1-bit measurements. The experimental setup follows the same framework as in the main body of the research, and the sampling technique is applied throughout.
+
+The results shown in Table 7 provide valuable insights into the performance of SIM-DMs under different parameter configurations. Specifically, we analyze the impact of hyperparameters  $C_s$ ,  $C_s'$ , and the number of function evaluations on model performance.
+
+When varying  $C_s$  while fixing  $C_s' = 55$  and  $\mathrm{NFE} = 50$ , we observe a clear upward trend in both PSNR and SSIM, indicating improved signal quality and better structural similarity with increasing  $C_s$ . However, LPIPS reaches its minimum value at a moderate  $C_s$ , suggesting that the best perceptual quality is achieved with balanced parameter settings. On the other hand, when  $C_s$  is fixed at 55 and  $C_s'$  is varied, we find that lower values of  $C_s'$  yield better performance across all metrics, with PSNR and SSIM reaching their peak at the lowest tested value. LPIPS remains relatively stable but shows a slight decrease at lower  $C_s'$  values. Finally, when we adjust the NFE with  $C_s = 55$  and  $C_s' = 55$ , all performance metrics show minimal fluctuation, suggesting that once NFE reaches a certain threshold, increasing it further does not yield significant improvements.
+
+In conclusion, these results underscore the flexibility of SIM-DMS with respect to parameter tuning. The parameter  $C_s$  demonstrates a clear impact on reconstruction quality, while lower values of  $C_s'$  tend to yield better results. Additionally, the stability across different NFE values indicates the model's efficiency even with fewer iterations. These findings provide valuable guidance for parameter selection in practical applications of SIM-DMS.
+
+Table 7. Ablation experiments for SIM-DMs on CIFAR-10 with  $m = {500}$  1-bit measurements,with different values of  ${C}_{s},{C}_{s}^{\prime }$  and NFEs. We mark bold the best scores, and underline the second best scores.  
+
+<table><tr><td rowspan="2">Value</td><td colspan="5">Cs(Cs&#x27; = 55)</td><td colspan="5">Cs&#x27;(Cs = 55)</td><td colspan="4">NFE (Cs = 55, C&#x27;s&#x27; = 55)</td></tr><tr><td>50</td><td>52</td><td>55</td><td>58</td><td>60</td><td>50</td><td>52</td><td>55</td><td>58</td><td>60</td><td>20</td><td>50</td><td>80</td><td>100</td></tr><tr><td>PSNR (↑)</td><td>14.441 ±1.037</td><td>14.625 ±1.119</td><td>14.860 ±1.201</td><td>14.956 ±1.275</td><td>14.971 ±1.334</td><td>15.337 ±1.354</td><td>15.214 ±1.252</td><td>14.860 ±1.201</td><td>14.451 ±1.185</td><td>14.205 ±1.146</td><td>14.860 ±1.201</td><td>14.860 ±1.201</td><td>14.860 ±1.201</td><td>14.860 ±1.201</td></tr><tr><td>SSIM (↑)</td><td>0.427 ±0.096</td><td>0.442 ±0.099</td><td>0.460 ±0.103</td><td>0.467 ±0.106</td><td>0.470 ±0.108</td><td>0.481 ±0.107</td><td>0.475 ±0.104</td><td>0.460 ±0.103</td><td>0.440 ±0.101</td><td>0.426 ±0.099</td><td>0.460 ±0.103</td><td>0.460 ±0.103</td><td>0.460 ±0.103</td><td>0.460 ±0.103</td></tr><tr><td>LPIPS (↓)</td><td>0.444 ±0.090</td><td>0.436 ±0.090</td><td>0.433 ±0.094</td><td>0.441 ±0.097</td><td>0.447 ±0.099</td><td>0.440 ±0.100</td><td>0.436 ±0.098</td><td>0.433 ±0.094</td><td>0.440 ±0.091</td><td>0.447 ±0.090</td><td>0.433 ±0.094</td><td>0.433 ±0.094</td><td>0.433 ±0.094</td><td>0.433 ±0.094</td></tr></table>
+
+# G. Ablation Study for SIM-DMIS
+
+In this section, we present a detailed analysis of the ablation experiments conducted on SIM-DMIS using the CIFAR-10 dataset with a measurement of  $m = 500$ . The experimental setup follows the same framework as in the main body of the research, and the DDIM sampling and DM2M inversion technique is applied throughout. The results are shown in Table 8.
+
+When varying  $C_s$  while fixing  $C_s' = 55$  and  $\mathrm{NFE} = 50$ , the trend of PSNR and SSIM are consistent with the SIM-DMS where higher values of  $C_s$  also led to better signal quality and structural similarity. On the other hand, when  $C_s$  is fixed at 55 and  $C_s'$  is varied, the trends for PSNR and SSIM are largely similar to the SIM-DMS results, where lower values of  $C_s'$  lead to better performance in these metrics. Interestingly, while the improvement in PSNR and SSIM is clear, LPIPS remains relatively stable across a wide range of  $C_s'$  values, with a slight decrease at lower values of  $C_s'$ , particularly around 50 and 52. Finally, when adjusting the NFE with  $C_s = 55$  and  $C_s' = 55$ , the results show minimal fluctuation in all performance metrics. This observation is consistent with the findings from SIM-DMS, where a similar behavior was noted.
+
+In summary, the findings from the ablation study of SIM-DMIS underline several important insights. The parameter  $C_s$  has a notable impact on both the signal quality and structural similarity of the reconstructed images, with higher values leading to better PSNR and SSIM. The parameter  $C_s'$  plays a crucial role in enhancing PSNR and SSIM, with lower values yielding better results, while LPIPS remains stable. Additionally, the minimal effect of increasing NFE suggests that SIM-DMIS is highly efficient and provides stable performance even with fewer evaluations.
+
+Table 8. Ablation experiments for SIM-DMIS on CIFAR-10 with  $m = {500}$  1-bit measurements. We mark bold the best scores, and underline the second best scores.  
+
+<table><tr><td rowspan="2">Value</td><td colspan="5">Cs(Cs&#x27; = 55, NFE=150)</td><td colspan="5">Cs&#x27;(Cs = 55, NFE=150)</td><td colspan="4">NFE (Cs = 55, C&#x27;s&#x27; = 55)</td></tr><tr><td>50</td><td>52</td><td>55</td><td>58</td><td>60</td><td>50</td><td>52</td><td>55</td><td>58</td><td>60</td><td>100</td><td>150</td><td>200</td><td>250</td></tr><tr><td rowspan="2">PSNR (↑)</td><td>15.388</td><td>15.562</td><td>15.733</td><td>15.809</td><td>15.820</td><td>16.247</td><td>16.091</td><td>15.733</td><td>15.322</td><td>15.057</td><td>15.733</td><td>15.733</td><td>15.733</td><td>15.733</td></tr><tr><td>±1.218</td><td>±1.331</td><td>±1.420</td><td>±1.513</td><td>±1.592</td><td>±1.655</td><td>±1.500</td><td>±1.420</td><td>±1.355</td><td>±1.307</td><td>±1.420</td><td>±1.420</td><td>±1.420</td><td>±1.420</td></tr><tr><td rowspan="2">SSIM (↑)</td><td>0.489</td><td>0.506</td><td>0.522</td><td>0.531</td><td>0.534</td><td>0.548</td><td>0.540</td><td>0.522</td><td>0.500</td><td>0.484</td><td>0.522</td><td>0.522</td><td>0.522</td><td>0.522</td></tr><tr><td>±0.097</td><td>±0.098</td><td>±0.099</td><td>±0.101</td><td>±0.104</td><td>±0.101</td><td>±0.099</td><td>±0.099</td><td>±0.099</td><td>±0.099</td><td>±0.099</td><td>±0.099</td><td>±0.099</td><td>±0.099</td></tr><tr><td rowspan="2">LPIPS (↓)</td><td>0.406</td><td>0.399</td><td>0.402</td><td>0.411</td><td>0.417</td><td>0.409</td><td>0.405</td><td>0.402</td><td>0.404</td><td>0.411</td><td>0.402</td><td>0.402</td><td>0.402</td><td>0.402</td></tr><tr><td>±0.095</td><td>±0.093</td><td>±0.095</td><td>±0.100</td><td>±0.102</td><td>±0.103</td><td>±0.098</td><td>±0.095</td><td>±0.093</td><td>±0.094</td><td>±0.095</td><td>±0.095</td><td>±0.095</td><td>±0.095</td></tr></table>
+
+# H. Comparative Evaluation of SIM-DMS and SIM-DMIS on Performance and Efficiency
+
+In this section, we evaluate the performance and efficiency of SIM-DMIS and SIM-DMIS on the FFHQ dataset. The experimental setup follows the same framework as in the main body of the research. The results presented in the main paper (Table 1) show that SIM-DMIS consistently outperforms QCS-SGM and SIM-DMFIS in both reconstruction quality and NFE. Notably, QCS-SGM requires over 11,000 NFEs to achieve competitive results, incurring a high computational cost. Due to the performance of SIM-DMIS over both methods, and the particularly high computational cost of QCS-SGM, we exclude QCS-SGM and SIM-DMFIS from the comparisons in this section.
+
+Performance Evaluation As discussed in Footnote 10 and Appendices F and G, ablation studies on CIFAR-10 show that further increasing the NFE for SIM-DMs does not lead to substantial performance enhancements.
+
+To further verify this behavior and to compare the relative performance of SIM-DMS and SIM-DMIS, we conduct additional experiments on the FFHQ dataset. The results, summarized in Table 9, indicate that under the same NFEs, SIM-DMIS consistently outperforms SIM-DMS across all standard metrics.
+
+Table 9. Quantitative results on FFHQ  $\left( {m = n/8 = {24576}}\right)$  . We mark bold the best scores, and underline the second best scores.  
+
+<table><tr><td>Method</td><td>NFE</td><td>PSNR (↑)</td><td>SSIM (↑)</td><td>LPIPS (↓)</td><td>FID (↓)</td></tr><tr><td>DPS-N</td><td>1000</td><td>11.14±1.46</td><td>0.37±0.09</td><td>0.69±0.05</td><td>349.24</td></tr><tr><td>DPS-L</td><td>1000</td><td>8.57±2.05</td><td>0.22±0.08</td><td>0.69±0.09</td><td>109.01</td></tr><tr><td>DAPS-N</td><td>1000</td><td>16.59±0.54</td><td>0.33±0.05</td><td>0.48±0.05</td><td>138.70</td></tr><tr><td>DAPS-L</td><td>1000</td><td>5.63±0.71</td><td>0.04±0.03</td><td>0.61±0.03</td><td>322.75</td></tr><tr><td>SIM- DMS</td><td>50</td><td>17.14±2.41</td><td>0.44±0.07</td><td>0.48±0.05</td><td>105.04</td></tr><tr><td>SIM- DMS</td><td>150</td><td>17.72±2.63</td><td>0.46±0.08</td><td>0.48±0.06</td><td>95.52</td></tr><tr><td>SIM-DMIS</td><td>50</td><td>18.78±3.09</td><td>0.58±0.10</td><td>0.41±0.08</td><td>89.47</td></tr><tr><td>SIM-DMIS</td><td>150</td><td>19.87±2.77</td><td>0.60±0.09</td><td>0.37±0.05</td><td>76.21</td></tr></table>
+
+Reconstruction Speed We conduct additional experiments to measure the reconstruction speed of SIM-DMS and SIM-DMIS. The inference time is averaged over 10 validation images from the FFHQ dataset. All experiments are executed on a single NVIDIA GeForce RTX 4090 GPU. As shown in Table 10, the results indicate that SIM-DMS and SIM-DMIS exhibit significantly faster reconstructions when compared to the competing methods.
+
+Table 10. Comparison of inference time (in seconds) for reconstructing 10 images on FFHQ. We mark bold the best scores, and underline the second best scores.  
+
+<table><tr><td>Method</td><td>NFE</td><td>Inference Time (s) (↓)</td></tr><tr><td>DPS-N</td><td>1000</td><td>142</td></tr><tr><td>DPS-L</td><td>1000</td><td>142</td></tr><tr><td>DAPS-N</td><td>1000</td><td>160</td></tr><tr><td>DAPS-L</td><td>1000</td><td>160</td></tr><tr><td>SIM- DMS</td><td>50</td><td>1.96</td></tr><tr><td>SIM-DMIS</td><td>150</td><td>5.66</td></tr></table>

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+# Learning Smooth and Expressive Interatomic Potentials for Physical Property Prediction
+
+Xiang Fu<sup>1</sup> Brandon M. Wood<sup>1</sup> Luis Barroso-Luque<sup>1</sup> Daniel S. Levine<sup>1</sup> Meng Gao<sup>1</sup> Misko Dzamba<sup>1</sup> C. Lawrence Zitnick<sup>1</sup>
+
+# Abstract
+
+Machine learning interatomic potentials (MLIPs) have become increasingly effective at approximating quantum mechanical calculations at a fraction of the computational cost. However, lower errors on held out test sets do not always translate to improved results on downstream physical property prediction tasks. In this paper, we propose testing MLIPs on their practical ability to conserve energy during molecular dynamic simulations. If passed, improved correlations are found between test errors and their performance on physical property prediction tasks. We identify choices which may lead to models failing this test, and use these observations to improve upon highly-expressive models. The resulting model, eSEN, provides state-of-the-art results on a range of physical property prediction tasks, including materials stability prediction, thermal conductivity prediction, and phonon calculations.
+
+# 1. Introduction
+
+Density Functional Theory (DFT), which models the electrons in materials and molecules, serves as the foundation for many modern drug and materials discovery workflows. Unfortunately, DFT calculations are notoriously computationally intensive, scaling cubically with the number of electrons in the system:  $O(n^{3})$ . Machine learning interatomic potentials (MLIPs) are promising in approximating and expediting DFT calculations. With increasing data set sizes and model innovations, MLIPs have shown substantial improvements in accuracy and generalization capabilities (Batatia et al., 2023; Merchant et al., 2023; Yang et al., 2024; Barroso-Luque et al., 2024).
+
+Fundamental AI Research (FAIR) at Meta. Correspondence to: Xiang Fu <xiangfu@meta.com>, C. Lawrence Zitnick <zitnick@meta.com>.
+
+Proceedings of the  $42^{nd}$  International Conference on Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).
+
+Predicting physical properties in chemistry and materials science often requires complex workflows involving numerous evaluations of DFT or MLIPs. For example, in molecular dynamics (MD) simulations, forces are predicted over thousands to millions of time steps. However, the MLIP literature has mostly focused on assessing models based on energy and force predictions over static DFT test sets rather than directly assessing their performance in complex simulations. This approach has limitations, as improved accuracy on test sets does not always lead to better predictions of physical properties (Póta et al., 2024; Loew et al., 2024).
+
+In this paper, we address two questions: Why does higher test accuracy sometimes fail to enhance a model's ability to predict physical properties, and how can we improve MLIPs to excel in this area? We first outline four critical property prediction tasks and identify the properties required for an MLIP to succeed in these tasks. These properties entail learning a conservative model with continuous and bounded energy derivatives, indicating a smoothly-varying and physically meaningful energy landscape. To test whether these properties hold, we propose testing the ability of MLIPs to practically conserve energy in MD simulations. We demonstrate models that pass this test have a higher correlation between test errors and property prediction accuracy.
+
+Building on these insights, we present a novel MLIP called eSEN and training approach that achieves state-of-the-art (SOTA) performance on complex property prediction tasks. Specifically, our model is capable of running energy-conserving MD simulations for out-of-distribution systems (Figure 1 (a)). For materials stability prediction, eSEN achieves a leading F1 score of 0.831 and a  $\kappa_{\mathrm{SRME}}$  of 0.340 on the compliant Matbench-Discovery benchmark (Riebesell et al., 2023; Pota et al., 2024). Previous models are only able to excel in one of these metrics (Figure 1 (b,c)). We also achieve a SOTA F1 score of 0.925 and  $\kappa_{\mathrm{SRME}}$  of 0.170 on the non-compliant category. On the MDR Phonon benchmark (Loew et al., 2024), SOTA results are found (Figure 1 (d)). Finally, eSEN achieves the highest test accuracy on the SPICE-MACE-OFF dataset (Kovacs et al., 2023).
+
+![](images/0add68526bf5748846ca85a4b3f54d5a95c7da46c36104cb323aef72e0d55fb3.jpg)  
+Figure 1. (a) Energy conservation in MD simulations. Direct-force models (Orb, eqV2) and CHGNet fail to conserve. (b) A higher F1 score on the Matbench-Discovery strongly correlates with a lower test-set energy MAE. (c) Test-set energy MAE and  $\kappa_{\mathrm{SRME}}$  on the Matbench-Discovery benchmark. (d) Test-set energy MAE and vibrational entropy MAE on the MDR Phonon benchmark. Our model (eSEN) achieves the best performance on all benchmarks. A higher correlation between test-set energy MAE and physical property prediction performance can be observed among energy-conserving models. All models are trained on MPTrj.
+
+# 2. Preliminaries
+
+# 2.1. Machine learning interatomic potentials
+
+Under the Born-Oppenheimer approximation (Oppenheimer, 1927) utilized by DFT (Parr et al., 1979), the Potential Energy Surface (PES) can be written as a function of positions,  $\pmb{r}$ , and atomic numbers,  $\pmb{a}$ :  $E(\pmb{r},\pmb{a})$ . Per-atom forces can be calculated by taking the negative gradient of the PES with respect to the atom positions,  $\pmb{F} = -\nabla_{\pmb{r}}E$ . For periodic systems such as inorganic materials, the lattice parameters  $\pmb{l}$  are also considered ( $E(\pmb{r},\pmb{a},\pmb{l})$ ), and the stress  $\sigma$  may also be calculated, which can be understood as the gradient of the potential energy surface with respect to the lattice parameters.
+
+The goal of an MLIP (Unke et al., 2021b) is to predict the exact same properties as DFT from a training dataset of DFT calculations (Chanussot et al., 2021; Riebesell et al., 2023; Loew et al., 2024). The most straightforward benchmark for MLIPs is to evaluate the model on a held-out test set of DFT calculations, and compare models based on the mean absolute error (MAE) or root mean squared error (RMSE) of
+
+energies, forces, or stresses. To bridge the gap between these performance metrics and practical applicability, we need to ensure they correlate with physical property prediction tasks, such as those described next.
+
+# 2.2. Physical property prediction tasks
+
+Geometry optimization/relaxation. Many computational chemistry and materials science tasks rely on atomic systems being in stable configurations, which correspond to minima of the PES. Stable states are found by minimizing the potential energy using an optimization procedure that iteratively updates atom positions based on the predicted forces  $(\boldsymbol{F} = -\nabla_{r}\boldsymbol{E})$ . Given that many physical properties are evaluated at or near equilibrium states, geometry optimization (also referred to as "relaxation") is usually the first step in most computational workflows.
+
+MD simulations. Simulating the time evolution of atomic systems enables us to gain understanding of various chemical and biological processes, as well as enabling the calculation of macroscopic properties, such as liquid densities, that can be experimentally verified. For the task of molecular dynamics simulation, we typically use a potential to compute the per-atom forces which are then used to numerically integrate Newton's equations of motion. In this work, we will focus on the microcanonical ensemble (NVE), where the number of particles (N), the volume of the system (V), and the energy of the system (E) are kept constant.
+
+Phonon and thermal conductivity calculations. Precise predictions of phonon band structures and vibrational modes are essential for understanding various material properties, including dynamical stability, thermal stability (Bartel, 2022; Fultz, 2010), thermal conductivity (Razeghi, 2002), and optoelectronic behavior (Ganose et al., 2021). The calculation of phonon band structures requires the MLIP to accurately predict higher-order derivatives and capture the subtle curvature of the true PES around critical points. Recent work (Póta et al., 2024) has demonstrated the usage of MLIPs in predicting thermal conductivity  $(\kappa)$  by solving the Wigner transport equation (Simoncelli et al., 2022). In order to accurately predict  $\kappa$ , MLIPs must reliably capture both harmonic and anharmonic phonon behavior, which necessitates the calculation of second and third derivatives of the learned PES.
+
+# 3. Desideratum for physical property prediction
+
+We begin the section by defining what it means for an MLIP to be energy conserving, which is a fundamental principle for applications such as MD simulations (Tuckerman, 2023). For many physical property prediction tasks that probe the higher-order derivatives of the PES it is also important that
+
+the PES's derivatives are well-behaved (they exist and are bounded). To indicate whether a PES meets these criteria, we discuss how an MLIP's ability to conserve energy given fixed simulation settings may be used.
+
+# 3.1. Conservative forces
+
+For a force model to be conservative, the work done by moving in a closed path must be zero, i.e., the integration of the forces along any path that starts and ends at the same point is zero:
+
+$$
+\oint \boldsymbol {F} \cdot d \boldsymbol {r} = 0 \tag {1}
+$$
+
+This property holds if the forces are calculated as the negative derivative of the PES with respect to the atom positions (Unke et al., 2021b). However, predicting forces as derivatives requires an additional backpropagation step through the network, which increases the computational cost of the MLIP. Alternatively, some networks (Liao et al., 2023; Neumann et al., 2024) directly predict forces using a separate force head to increase efficiency<sup>1</sup>. Although direct-force models can achieve high accuracy, their non-conservative nature leads to significantly larger errors in certain property prediction tasks (Fu et al., 2023; Loew et al., 2024; Póta et al., 2024; Bigi et al., 2024).
+
+# 3.2. Bounded energy derivatives
+
+Conservative forces is a necessary but not sufficient condition for an MLIP to demonstrate energy conservation in MD. In practice, MD simulations use a finite-order numerical integration algorithm and a finite time step  $\Delta t$ , which introduces truncation errors. The most commonly used integrator for the NVE ensemble is the Verlet algorithm—a second-order integrator. The Verlet integrator is known to approximately conserve the total energy of the system in long-time simulations. As shown by Theorem 5.1 of Hairer et al. 2003, the total energy drift of a simulation satisfies
+
+$$
+| E (\pmb {r} _ {T}, \pmb {a}) - E (\pmb {r} _ {0}, \pmb {a}) | \leq C \Delta t ^ {2} + C _ {N} \Delta t ^ {N} T, (2)
+$$
+
+where  $T, 0 \leq T \leq \Delta t^{-N}$ , is the total simulation time,  $N$  is a positive integer representing the highest order for which the  $N$ th-order derivative of  $E$  is continuously differentiable with a bounded derivative, and  $\boldsymbol{r}_0$  and  $\boldsymbol{r}_T$  are the starting and ending positions of the atoms in the simulation respectively. The constants  $C$  and  $C_N$  are independent of  $T$  and  $\Delta t$ . The energy drift bound contains two terms: the first term represents a time-independent fluctuation of  $O(\Delta t^2)$  and the second term represents the long-term energy conservation. The proof for this theorem is long and technical, for
+
+![](images/3a94b44fe2dbc563755dbcc745fdb9e8920663e223fafb13cc064667d2041ed0.jpg)  
+Figure 2. (a) The eSEN architecture. The high-level architecture is similar to Transformer/Equiformer, while the edgewise/nodewise layers are simplified/enhanced. The final-layer  $L = 0$  features are used to predict nodewise energy, which is summed to get the total potential energy  $E$ . Forces and stress are obtained through back-propagation. (b) The Edgewise Convolution layer in eSEN.
+
+![](images/8e7267438b0687ea9399c43d356aef0f55296af16fb93809b051d267c843823b.jpg)
+
+which we refer interested readers to Hairer et al. 2003 and Hairer et al. 2006 for more details.
+
+In Equation (2), the  $\Delta t^{N}$  in the second term and the bound on the simulation time  $T\leq \Delta t^{-N}$  implies that the PES must be continuously differentiable to high order for energy conservation in long-time simulations. The critical constant  $C_N$  depends on the bounds of the derivatives of  $E$  up to the  $(N + 1)\mathrm{th}$  order. This implies that, given a fixed time step size,  $E$  and its higher-order derivatives up to the  $(N)$ th order all need to be continuously differentiable with bounded derivatives to maintain long-time conservation. If the derivatives of a PES are more tightly bound, approximate energy conservation will be maintained even at larger step sizes  $\Delta t$ . Therefore, the magnitude of  $\Delta t$  for which the energy is stable can be viewed as a proxy for the derivative bounds of the estimated PES. Alternatively, if a certain time step is known to be stable when using DFT, we can determine whether an MLIP has similar bounds on higher-order derivatives by testing whether it is also stable using the same time step.
+
+# 4.eSEN
+
+We propose equivariant Smooth Energy Network (eSEN), a new MLIP architecture that improves upon architectures that demonstrate high test accuracies to achieve effective physical property predictions. eSEN is a message-passing neural network that conducts multiple blocks of edgewise
+
+and nodewise neural processing. Initially, all nodes are embedded as multi-channel spherical harmonic representations. Each eSEN layer block updates the node embedding by conducting an edgewise convolution, followed by a node-wise feed-forward network with normalization layers and residual connections between all layers.
+
+A model diagram is shown in Figure 2. eSEN utilizes the same SO2 convolution layer from the equivariant spherical channel network (eSCN) architecture (Passaro & Zitnick, 2023) inside the edgewise convolution block. Compared to eSCN, our edgewise convolution blocks first concatenate the source and target node embedding, then apply two SO2 convolution layers with an intermediate non-linearity. We also add an envelope function (details in Section 5) which is not in eSCN. The nodewise feed-forward layer uses two equivariant linear layers and an intermediate SiLU-based gated non-linearity (Weiler et al., 2018; Geiger & Smidt, 2022), which is the same as Equiformer (Liao & Smidt, 2022). Unlike eSCN and EquiformerV2 (Liao et al., 2023), which projects the spherical-harmonics channels onto spatial grids for nodewise processing, the nodewise layers in eSEN do not discretize the node representations. As we demonstrate in Section 5, this design improves the ability of the model to conserve energy. Normalization is performed using the equivariant layer normalization (Ba, 2016) proposed by Equiformer (Liao & Smidt, 2022). In the next section, we conduct an in-depth analysis of the key design choices for energy conservation, which we argue is important for accurate physical property prediction.
+
+# 5. Design choices for enhancing physical property prediction
+
+As discussed in Section 3, having conservative forces with continuous and bounded energy derivatives are properties an MLIP should obey for MD simulations. It can also be seen as a prerequisite for the MLIP to accurately capture higher-order behavior of the PES and thus high accuracy in physical property prediction tasks such as phonon calculations. Motivated by this observation, we identify design choices that impact a model's ability to conserve energy and whether its PES varies smoothly. These design choices can be categorized into three aspects: (1) conservative vs. direct-force prediction; (2) discretization of the representation; and (3) obtaining a continuous and smoothly varying PES. For many of these design aspects, their impact on the desired properties is not well understood.
+
+To quantify whether an MLIP's PES is continuous and smoothly varying, we measure the ability of the resulting MLIP to conserve energy during MD simulations with a predetermined fixed time step. We trained eSEN models under the same hyperparameters while ablating one design choice at a time. We construct out-of-distribution (OOD) MD
+
+simulation tasks for both inorganic materials and organic molecules using models trained on the MPTrj (Jain et al., 2013; Deng et al., 2023) and the SPICE-MACE-OFF (Eastman et al., 2023; Kovacs et al., 2023) datasets. For inorganic materials, we compute an average conservation error over 81 NVE MD simulations of 100 ps based on the TM23 dataset's simulation settings (Owen et al., 2024). For organic molecules, we compute an average conservation error over 7 NVE MD simulations of 100 ps based on the MD22 dataset's simulation settings (Chmiela et al., 2023). All eSEN models are 2-layer with 3.2M trainable parameters. We include details regarding the task protocol in Appendix A.
+
+# 5.1. Direct-force prediction
+
+Models that directly predict forces  $\hat{F}$  from the atomic configuration may produce forces that are inconsistent with the energy prediction, i.e.,  $\hat{F} \neq -\nabla_{r}\hat{E}$ , and more importantly are unlikely to be conservative. From the perspective of minimizing the test error, the direct-force approach has strong motivations: it avoids the backward pass for force prediction, which significantly improves model efficiency and enables low-precision training which further accelerates training. Empirically, current SOTA accuracy on the OC20, OC22, and Matbench-Discovery (Chanussot et al., 2021; Tran et al., 2023; Riebesell et al., 2023) benchmarks are achieved by direct-force models. Despite this, the direct-force formulation results in significant energy drift in MD simulations, as shown in Figure 4 (a1, a2). For this reason, we compute forces as the negative gradient of the PES with respect to the atom positions in eSEN.
+
+![](images/14bce4b453c9d6063e78ad3cf675c3639395e4f0308a80f70fa58dc011f2900a.jpg)  
+Figure 3. Validation loss curves for epoch and wallclock time.
+
+Direct-force pre-training. Although direct-force models are not suitable for certain physical property prediction tasks, they may still offer advantages (Bigi et al., 2024; Amin et al., 2025). We demonstrate their efficiency can offer significant benefit as a pre-training strategy for a conservative model. Figure 3 shows the validation loss of 2-layer eSEN models trained on the MPTrj dataset: direct-force, conservative, and conservative fine-tuning from a pre-trained direct-force backbone. We start from a direct-force model trained for 60 epochs, remove its direct-force prediction head, and fine-
+
+![](images/6d66b0f41773e96ebf564fbdd061a48c781b7767e14310395ab148948a12eb39.jpg)
+
+![](images/840ddf741a2c35363ccca783188e13affdf4123624c83d858b9c7bf02b068403.jpg)
+
+![](images/6c47773461827b4b5f34ce2140cdd3476fef4c33aebbf44e6c662cc46687702a.jpg)
+
+![](images/b9e6bfdaa03d66a0ea506ae41937fec54e781dc5abe7f0f6e8eb68ba25e5aa72.jpg)  
+Figure 4. Conservation error on the TM23 task (top row) and MD22 task (bottom row) for ablating design choices of eSEN. Models that conserve energy are bolded in the legends.
+
+![](images/6a537e8a5fb762ce7a0bdc8f1aeed56772de0e0ed91104e3c47eb9c253cbc173.jpg)
+
+![](images/c969f06d48c12b1081d3f333580e1f5ef48cbf2315faf5b4f4a52d2c54b807e5.jpg)
+
+tune using conservative force prediction. The conservative fine-tuned model achieves a lower validation loss after being trained for 40 epochs compared to the from-scratch conservative model being trained for 100 epochs. The fine-tuning strategy also reduces the wallclock time for model training by  $40\%$ . The strategy of combining direct-force pre-training and conservative fine-tuning is also shown to be effective under different data/model settings (Bigi et al., 2024).
+
+# 5.2. Representation discretization
+
+As proposed by Cohen & Welling 2016 and later used in eSCN and EquiformerV2 (Zitnick et al., 2022; Passaro & Zitnick, 2023; Liao et al., 2023), non-linearities may be performed by projecting the spherical harmonics to a discrete grid. A  $1 \times 1$  convolution or pointwise non-linearity may then be applied to this grid, which then get projected back to the spherical-harmonics space. The non-linear step may introduce higher-frequency signals than cannot be properly represented by the spherical harmonics, i.e., they are beyond the Nyquist frequency. This can lead to sampling errors that break strict equivariance and energy conservation. This problem can be mitigated by sampling the grid at higher resolutions as shown in Figure 4 (b1, b2). In eSEN, we instead use the SiLU-based equivariant Gated non-linearity (Weiler et al., 2018; Geiger & Smidt, 2022) that performs the nonlinearity directly in the spherical harmonic representation. This does not require a projection to a discrete grid, so the model is perfectly equivariant and conservative up to numerical accuracy.
+
+# 5.3. Smoothly varying PES
+
+Subtle choices in the design of MLIPs can have a significant impact on whether a PES varies smoothly and can even lead to the presence of discontinuities. These include how neighboring atoms are chosen, whether envelope functions are used near atom distance cutoffs, and which basis functions are used to embed pairwise atom distances. We discuss each of these in turn.
+
+A maximum number of neighbors limit in graph construction has been found to improve training efficiency without compromising test error (Liao et al., 2023; Qu & Krishnapriyan, 2024). However, it results in a discontinuity in the learned PES as the nearest-K neighbors may change drastically under a small perturbation of the atom positions. As shown in Figure 4 (c1, c2), having a maximum neighbor limit breaks energy conservation. In eSEN, instead of limiting the number of neighbors, we use the common approach of applying a distance cutoff (6Å) under which all neighbors are kept.
+
+Envelope functions were first introduced in the DimeNet architecture (Gasteiger et al., 2020a) to improve model smoothness. The radial basis function used in MLIPs is not twice continuously differentiable due to the use of a finite cutoff during graph construction. By applying a polynomial envelope function on the edge messages, the values in an edge message and its first/higher-order derivatives with respect to atom positions decays to 0 when the edge distance approaches the cutoff distance. Figure 4 (c1, c2)
+
+Table 1. Test set MAE for design choices studied in Section 3. The conserved model significantly outperforms the direct-force model on SPICE-MACE-OFF.  $N_{\mathrm{basis}} = 512$  performs slightly better on SPICE but slightly worse on MPTrj. Other eSEN variants all have similar test errors on MPTrj/SPICE-MACE-OFF. Energy MAE is in meV/atom. Force MAE is in meV/Å. Stress MAE is in meV/Å/atom.  
+
+<table><tr><td rowspan="2">Model</td><td colspan="3">MPTrj</td><td colspan="2">SPICE</td></tr><tr><td>Energy</td><td>Force</td><td>Stress</td><td>Energy</td><td>Force</td></tr><tr><td>eSEN</td><td>17.02</td><td>43.96</td><td>0.14</td><td>0.23</td><td>6.36</td></tr><tr><td>eSEN, direct</td><td>18.66</td><td>43.62</td><td>0.16</td><td>0.56</td><td>10.98</td></tr><tr><td>eSEN, neighbor limit</td><td>17.30</td><td>44.11</td><td>0.14</td><td>0.24</td><td>6.52</td></tr><tr><td>eSEN, no envelope</td><td>17.60</td><td>44.69</td><td>0.14</td><td>0.23</td><td>6.33</td></tr><tr><td>eSEN, Nbasis=512</td><td>19.87</td><td>48.29</td><td>0.15</td><td>0.19</td><td>5.40</td></tr><tr><td>eSEN, Bessel</td><td>17.65</td><td>44.83</td><td>0.15</td><td>0.20</td><td>5.54</td></tr><tr><td>eSEN, discrete, res=6</td><td>17.05</td><td>43.10</td><td>0.14</td><td>0.26</td><td>6.34</td></tr><tr><td>eSEN, discrete, res=10</td><td>17.11</td><td>43.13</td><td>0.14</td><td>0.33</td><td>6.57</td></tr><tr><td>eSEN, discrete, res=14</td><td>17.12</td><td>43.09</td><td>0.14</td><td>0.33</td><td>6.51</td></tr></table>
+
+shows a model fails to conserve energy without the envelope function.
+
+Radial basis functions are commonly used to embed interatomic distances (Bartók et al., 2013). A larger number of basis functions (512 in Passaro & Zitnick 2023, as opposed to 10 in eSEN's default setting) allows higher-frequency signals to pass through the network. This can lead to the PES being more sensitive to small shifts in the atom positions. In our experiments, using a large number of basis functions breaks conservation for the TM23 tasks, but is able to conserve energy for the MD22 tasks. Using a Bessel radial basis function (as opposed to a Gaussian radial basis in the default setting) does not impact conservation properties in both tasks.
+
+# 5.4. Ablation studies
+
+Many of the architecture choices described above have negligible impact on the test set errors as shown in Table 1. However, as shown in Figure 4, they can have a dramatic impact on whether a model is conservative in practice. If a model is found to be conservative, stronger correlations are found between test errors and property prediction tasks (Figure 1 and Figure 6).
+
+# 6. Experiments
+
+In the previous section, we demonstrated the design of eSEN results in its ability to be energy-conserving in MD simulations. In this section, we evaluate eSEN in physical property prediction tasks: (1) materials stability prediction based on geometry optimization; (2) thermal conductivity prediction; and (3) phonon calculation. We also demonstrate the correlation between test energy MAE and physical property prediction tasks for eSEN.
+
+# 6.1. Matbench Discovery
+
+The Matbench-Discovery benchmark evaluates a model's ability to predict ground-state (0 K) thermodynamic stability through geometry optimization and energy prediction. It is a widely used benchmark for evaluating ML models in materials discovery. The compliant benchmark only includes models trained on the MPTrj (Jain et al., 2013; Deng et al., 2023) dataset or its subset, which facilitate a fair comparison of model architectures. The F1 score is the primary metric used to rank models. We train an eSEN with 30M parameters on MPTrj for 60 epochs of direct-force pre-training and 40 epochs of conservative fine-tuning. DeNS (Liao et al., 2024) is used during direct-force pre-training. As shown in Table 2, eSEN-30M-MP achieves an F1 score of 0.831—the highest among all compliant models. eSEN-30M-MP also achieves the lowest root mean square deviation (RMSD) when comparing the relaxed structures to the ground truth DFT reference.
+
+The thermal conductivity prediction task requires accurate modeling of harmonic and anharmonic phonons in materials, which tests the accuracy of second and third order derivatives of the learned PES. The primary metric is the symmetric relative mean error in predicting thermal conductivity ( $\kappa_{\mathrm{SRME}}$ ). We follow the protocol set forth in the Matbench-Discovery benchmark (Riebesell et al., 2023; Póta et al., 2024) to predict thermal conductivity  $\kappa$ . After running a structural relaxation,  $\kappa$  is computed using second and third order force constants obtained from phonon calculations using the supercell method.
+
+As shown in Table 2, our model achieves a  $\kappa_{\mathrm{SRME}}$  of 0.340 under the default evaluation protocol proposed by Póta et al. 2024. Notably, our model excels in both the F1 score and  $\kappa_{\mathrm{SRME}}$ , while all previous models only achieve SOTA performance on one or the other of these metrics.
+
+The non-compliant Matbench-Discovery benchmark includes models trained on datasets other than MPTrj. eSEN30M-OAM is an eSEN model with 30 million parameters pre-trained on the OMat24 (Barroso-Luque et al., 2024) dataset then fine-tuned on the subsampled Alexandria (sAlex) dataset (Barroso-Luque et al., 2024; Schmidt et al., 2024) and MPTrj dataset. As shown in Table 3, eSEN30M-OAM achieves the best performance among all noncompliant models with an F1 score of 0.925, a  $\kappa_{\mathrm{SRME}}$  of 0.170, and an RMSD of 0.0608, significantly advancing state-of-the-art.
+
+# 6.2. MDR phonon benchmark
+
+The MDR Phonon benchmark (Loew et al., 2024) assesses the performance of MLIPs in predicting key phonon properties, including maximum phonon frequency ( $\omega_{\mathrm{max}}$ ), entropy ( $S$ ), free energy ( $F$ ) and heat capacity at constant volume
+
+Table 2. Matbench-Discovery benchmark results of compliant models (trained only on MPtrj or its subset) with results on the unique prototype split. MAE is in units of eV/atom.  $(\uparrow/\downarrow)$  stands for higher/lower the better.  
+
+<table><tr><td>Metric</td><td>eSEN-30M-MP</td><td>eqV2 S DeNS</td><td>MatRIS-MP</td><td>AlphaNet-MP</td><td>DPA3-v2-MP</td><td>ORB v2 MPtrj</td><td>SevenNet-13i5</td><td>GRACE-2L-MPtrj</td><td>MACE-MP-0</td><td>CHGNet</td><td>M3GNet</td></tr><tr><td>F1 ↑</td><td>0.831</td><td>0.815</td><td>0.809</td><td>0.799</td><td>0.786</td><td>0.765</td><td>0.760</td><td>0.691</td><td>0.669</td><td>0.613</td><td>0.569</td></tr><tr><td>DAF ↑</td><td>5.260</td><td>5.042</td><td>5.049</td><td>4.863</td><td>4.822</td><td>4.702</td><td>4.629</td><td>4.163</td><td>3.777</td><td>3.361</td><td>2.882</td></tr><tr><td>Precision ↑</td><td>0.804</td><td>0.771</td><td>0.772</td><td>0.743</td><td>0.737</td><td>0.719</td><td>0.708</td><td>0.636</td><td>0.577</td><td>0.514</td><td>0.441</td></tr><tr><td>Accuracy ↑</td><td>0.946</td><td>0.941</td><td>0.938</td><td>0.933</td><td>0.929</td><td>0.922</td><td>0.920</td><td>0.896</td><td>0.878</td><td>0.851</td><td>0.813</td></tr><tr><td>MAE ↓</td><td>0.033</td><td>0.036</td><td>0.037</td><td>0.041</td><td>0.039</td><td>0.045</td><td>0.044</td><td>0.052</td><td>0.057</td><td>0.063</td><td>0.075</td></tr><tr><td>R2 ↑</td><td>0.822</td><td>0.788</td><td>0.803</td><td>0.745</td><td>0.804</td><td>0.756</td><td>0.776</td><td>0.741</td><td>0.697</td><td>0.689</td><td>0.585</td></tr><tr><td>κSRME ↓</td><td>0.340</td><td>1.676</td><td>0.861</td><td>1.31</td><td>0.959</td><td>1.725</td><td>0.550</td><td>0.525</td><td>0.647</td><td>1.717</td><td>1.412</td></tr><tr><td>RMSD ↓</td><td>0.0752</td><td>0.0757</td><td>0.0773</td><td>0.1067</td><td>0.0823</td><td>0.1007</td><td>0.0847</td><td>0.0897</td><td>0.0915</td><td>0.0949</td><td>0.1117</td></tr></table>
+
+Table 3. Matbench-Discovery benchmark results of non-compliant models with results on the unique prototype split.  
+
+<table><tr><td>Model</td><td>eSEN-30M-OAM</td><td>eqV2-M-OAM</td><td>ORB v3</td><td>SevenNet-MF-ompa</td><td>DPA3-v2-OpenLAM</td><td>GRACE-2L-OAM</td><td>MatterSim-v1-5M</td><td>MACE-MPA-0</td><td>GNoME</td></tr><tr><td>F1 ↑</td><td>0.925</td><td>0.917</td><td>0.905</td><td>0.901</td><td>0.890</td><td>0.880</td><td>0.862</td><td>0.852</td><td>0.829</td></tr><tr><td>DAF ↑</td><td>6.069</td><td>6.047</td><td>5.912</td><td>5.825</td><td>5.747</td><td>5.774</td><td>5.852</td><td>5.582</td><td>5.523</td></tr><tr><td>Precision ↑</td><td>0.928</td><td>0.924</td><td>0.904</td><td>0.879</td><td>0.879</td><td>0.883</td><td>0.895</td><td>0.853</td><td>0.844</td></tr><tr><td>Accuracy ↑</td><td>0.977</td><td>0.975</td><td>0.971</td><td>0.969</td><td>0.966</td><td>0.963</td><td>0.959</td><td>0.954</td><td>0.955</td></tr><tr><td>MAE ↓</td><td>0.018</td><td>0.020</td><td>0.024</td><td>0.021</td><td>0.022</td><td>0.023</td><td>0.024</td><td>0.028</td><td>0.035</td></tr><tr><td>R2 ↑</td><td>0.866</td><td>0.848</td><td>0.821</td><td>0.867</td><td>0.869</td><td>0.862</td><td>0.863</td><td>0.842</td><td>0.785</td></tr><tr><td>\( \kappa_{\text{SRME}} \)↓</td><td>0.170</td><td>1.771</td><td>0.210</td><td>0.317</td><td>0.687</td><td>0.294</td><td>0.574</td><td>0.412</td><td>N/A</td></tr><tr><td>RMSD ↓</td><td>0.0608</td><td>0.0691</td><td>0.0750</td><td>0.0639</td><td>0.0679</td><td>0.0666</td><td>0.0733</td><td>0.0731</td><td>N/A</td></tr></table>
+
+$(C_V)$ , for around 10,000 materials. The evaluation follows the testing protocol outlined by Loew et al. 2024. Section 6.2 shows the resulting MAE of our model and those of several other models<sup>2</sup>. eSEN achieves SOTA results in both compliant and non-compliant categories.
+
+Our results are consistent with those reported by Loew et al. 2024, showing that conservative MLIPs significantly outperform direct-force models in terms of prediction accuracy when tested using phonon calculations with a displacement of  $0.01\AA$ . The high error of direct-force models can be largely attributed to high-frequency prediction errors at small displacements (Loew et al., 2024). Increasing the displacement used in the finite-difference phonon calculations to  $0.2\AA$  can considerably improve prediction accuracy of direct-force models (with caveats). We include a more detailed analysis of the relationship between atom displacement and phonon prediction in Appendix B.
+
+In physical phonon calculations, we expect the results to converge as the displacement goes to zero. By examining the resulting phonon band structure, we can gain insight into this behavior. Figure 5 presents the predicted phonon band structure and density of states for three representative materials using eSEN. The predicted phonon bands exhibit convergence as the displacement decreases. In contrast, Figures Figure C.11 and Figure C.12 display the phonon bands for the same three materials predicted using eqV2-S-DeNS (direct-forces), which not only fail to demonstrate convergence but also exhibit significant errors, including missing
+
+Table 4. Summary of model performance on the MDR Phonon benchmark. Metrics include maximum phonon frequency (MAE( $\omega_{\text{max}}$ ), in K), the vibrational entropy (MAE(S), in J/K/mol), the Helmholtz free energy (MAE(F), in kJ/mol), and the heat capacity at constant volume (MAE( $C_V$ ), in J/K/mol). *The OMat24 dataset uses a slightly different DFT setting from the DFT setting of the MDR Phonon benchmark.
+
+<table><tr><td>Compliant models</td><td>MAE(ωmax)</td><td>MAE(S)</td><td>MAE(F)</td><td>MAE(CV)</td></tr><tr><td>M3GNet</td><td>98</td><td>150</td><td>56</td><td>22</td></tr><tr><td>CHGNet</td><td>89</td><td>114</td><td>45</td><td>21</td></tr><tr><td>MACE</td><td>61</td><td>60</td><td>24</td><td>13</td></tr><tr><td>GRACE-2L (r6)</td><td>40</td><td>25</td><td>9</td><td>5</td></tr><tr><td>SevenNet-0</td><td>40</td><td>48</td><td>19</td><td>9</td></tr><tr><td>SevenNet-13i5</td><td>26</td><td>28</td><td>10</td><td>5</td></tr><tr><td>eSEN-30M-MP</td><td>21</td><td>13</td><td>5</td><td>4</td></tr><tr><td colspan="5">Direct-force models</td></tr><tr><td>Orb MPTrj [0.01 Å]</td><td>309</td><td>476</td><td>64</td><td>181</td></tr><tr><td>Orb MPTrj [0.2 Å]</td><td>61</td><td>34</td><td>11</td><td>8</td></tr><tr><td>eqV2-S-DeNS [0.01 Å]</td><td>280</td><td>224</td><td>54</td><td>94</td></tr><tr><td>eqV2-S-DeNS [0.2 Å]</td><td>58</td><td>26</td><td>8</td><td>8</td></tr><tr><td colspan="5">Non-compliant models</td></tr><tr><td>eqV2-M-OAM [0.01 Å]</td><td>780</td><td>403</td><td>241</td><td>100</td></tr><tr><td>eqV2-M-OAM [0.2 Å]</td><td>50</td><td>25</td><td>7</td><td>9</td></tr><tr><td>MatterSim</td><td>17</td><td>15</td><td>5</td><td>3</td></tr><tr><td>GRACE-2L-OAM</td><td>19</td><td>14</td><td>5</td><td>4</td></tr><tr><td>SevenNet-MF-ompa</td><td>15</td><td>8</td><td>3</td><td>3</td></tr><tr><td>eSEN-30M-OAM</td><td>15</td><td>10</td><td>4</td><td>3</td></tr></table>
+
+acoustic branches and spurious imaginary frequencies.
+
+While we find that the OMat-trained models (without sAlex/MPTrj finetuning) may provide a lower error on phonon prediction (7/7/2/2 for  $\omega_{\mathrm{max}} / S / F / C_V$  MAE), we refrain from direct comparison due to mismatch in level of theory. We attribute this result to the softening issue of the sAlex and MPTrj dataset (Deng et al., 2025; Barroso-Luque et al., 2024), which OMat24 addresses. We refer interested
+
+![](images/6b08f218d03ef6d8204faf5503319828d20c566d4ddbed451d6701e83cab305b.jpg)  
+Figure 5. Predicted phonon band structure and density of states (DOS) of Si (diamond structure), CsCl (CsCl structure), AlN (wurtzite structure) using eSEN at different displacement values. DFT baseline is taken from the PBE MDR dataset (Loew et al., 2024) calculated using a displacement of  $0.01\AA$
+
+![](images/167d0497ed02dac060f1b4aec7910fb00875b50a7f67975b7f1e558a24927179.jpg)
+
+![](images/6ed6abebb45516113bbd7af0a74429e21f8d0b294c244d25d0ebbc25bf650913.jpg)
+
+Table 5. Test set MAE for SPICE-MACE-OFF. Energy (E) MAE is in meV/atom. Force (F) MAE is in meV/Å. *EscAIP-45M is a direct-force model.  
+
+<table><tr><td rowspan="2">Dataset</td><td colspan="2">MACE-4.7M</td><td colspan="2">ESCaIP-45M*</td><td colspan="2">eSEN-3.2M</td><td colspan="2">eSEN-6.5M</td></tr><tr><td>E</td><td>F</td><td>E</td><td>F</td><td>E</td><td>F</td><td>E</td><td>F</td></tr><tr><td>PubChem</td><td>0.88</td><td>14.75</td><td>0.53</td><td>5.86</td><td>0.22</td><td>6.10</td><td>0.15</td><td>4.21</td></tr><tr><td>DES370K M.</td><td>0.59</td><td>6.58</td><td>0.41</td><td>3.48</td><td>0.17</td><td>1.85</td><td>0.13</td><td>1.24</td></tr><tr><td>DES370K D.</td><td>0.54</td><td>6.62</td><td>0.38</td><td>2.18</td><td>0.20</td><td>2.77</td><td>0.15</td><td>2.12</td></tr><tr><td>Dipeptides</td><td>0.42</td><td>10.19</td><td>0.31</td><td>5.21</td><td>0.10</td><td>3.04</td><td>0.07</td><td>2.00</td></tr><tr><td>Sol. AA</td><td>0.98</td><td>19.43</td><td>0.61</td><td>11.52</td><td>0.30</td><td>5.76</td><td>0.25</td><td>3.68</td></tr><tr><td>Water</td><td>0.83</td><td>13.57</td><td>0.72</td><td>10.31</td><td>0.24</td><td>3.88</td><td>0.15</td><td>2.50</td></tr><tr><td>QMugs</td><td>0.45</td><td>16.93</td><td>0.41</td><td>8.74</td><td>0.16</td><td>5.70</td><td>0.12</td><td>3.78</td></tr></table>
+
+readers to Deng et al. 2025 and Barroso-Luque et al. 2024 for a detailed discussion on the softening issue of some DFT datasets and its relation to phonon properties.
+
+# 6.3. SPICE-MACE-OFF
+
+We train and evaluate eSEN models on the SPICE-MACE-OFF dataset (Kovács et al., 2023), which is built upon the SPICE dataset (Eastman et al., 2023). As shown in Table 5, eSEN with 6.5M parameters outperforms MACE-OFF-L (4.7M parameters) and EscAIP (45M parameters, direct-force) on all test-set splits for both energy and force MAE. We also include results for eSEN with 3.2M parameters, which has inference efficiency similar to MACE-4.7M, while achieving lower test energy/force MAE. More details on the inference efficiency benchmark are included in Appendix C.
+
+# 6.4. Test-set error for model development
+
+In Figure 1, we showed the correlation between test error and physical property prediction tasks for different architectures. Figure 6 demonstrates this correlation for different variants of eSEN (with a 1k-materials subset of the MDR Phonon benchmark for efficiency). In particular, among models that pass the MD energy conservation test, a strong
+
+Figure 6. Test error correlation across several property prediction tasks for eSEN variants. Conservative models are shown as boxes and those found to not conserve as crosses. Note metrics for conservative models have a stronger correlation with test set errors.  
+![](images/36c202e7d97e5088549be3092b956fed5e77e0d0f8f4a62a2a39edb5ed5ad42d.jpg)  
+eSEN-30M eSEN, Bessel eSEN, neighbor limit eSEN-6.5M eSEN, direct-force eSEN, no envelope
+
+![](images/e6c19ad3c5e2bd8d9ba794f94d24c44fb395ff441321bc51d90e45e23be76668.jpg)  
+eSEN-3.2M eSEN, discrete eSEN,  $N_{\mathrm{basis}} = 512$
+
+correlation between test error and  $\kappa_{\mathrm{SRME}}$ /vibrational entropy MAE can be observed. We include experimental details about Figure 1 and Figure 6 in Appendix A and additional results for other phonon properties in Appendix B.
+
+# 7. Related works
+
+MLIP architectures have made significant progress since their initial proposal (Behler & Parrinello, 2007). These architectures are usually symmetry-preserving (Smith et al., 2017; Schutt et al., 2017; Gilmer et al., 2017; Chmiela et al., 2017; Artrith et al., 2017; Unke & Meuwly, 2018; Zhang et al., 2018; Zubatyuk et al., 2019; Smith et al., 2020; Kovacs et al., 2021), with increasingly expressive atom environment embeddings and message-passing operations (Gasteiger et al., 2020b; 2021; Schutt et al., 2021; Liu et al., 2021; Unke et al., 2021a; Chen & Ong, 2022; Deng et al., 2023; Cheng, 2024; Yin et al., 2025). Notably, equiv-
+
+ariant architectures based on spherical harmonics representations (Thomas et al., 2018; Thölke & De Fabritiis, 2021; Batzner et al., 2022; Musaelian et al., 2022; Batatia et al., 2022; Passaro & Zitnick, 2023; Liao et al., 2023; Bochkarev et al., 2024; Park et al., 2024; Batatia et al., 2025) have shown strong performance on large-scale datasets. Meanwhile, the high computational cost of these architectures has sparked significant interest in scalable architectures that may not respect physical principles such as energy conservation (Langer et al., 2024; Brehmer et al., 2024; Hu et al., 2021; Yang et al., 2024; Qu & Krishnapriyan, 2024; Neumann et al., 2024; Rhodes et al., 2025). These models have demonstrated strong performance in accuracy, scalability, and relaxation tasks (Chanussot et al., 2021; Riebesell et al., 2023). While their non-physical nature may make them unsuitable for direct usage in some physical property prediction tasks, they may still provide benefit by using the pre-training strategy proposed in this paper and Bigi et al. 2024, distilling them to conservative models (Amin et al., 2025), or combining them with a conservative model using multiple-time-step integration (Bigi et al., 2024).
+
+MLIPs and physical observables. While MLIPs continue to improve, it is necessary to evaluate them in realistic tasks that are relevant to scientific discovery. Physical property prediction benchmarks that involve geometry optimization (Riebesell et al., 2023; Lan et al., 2023; Wander et al., 2024), MD simulations (Fu et al., 2023; Kovacs et al., 2023; Moore et al., 2024; Sabanes Zariquiey et al., 2024; Eastman et al., 2024), vibrational analysis and phonon calculations (Pota et al., 2024; Loew et al., 2024; Wines & Choudhary, 2024), and others are increasing in scale with broader applications and wider adoption. Training strategies for learning from physical observables (Wang et al., 2020; Greener, 2024; Röcken et al., 2024; Raja et al., 2024) and the higher-order derivatives of the PES (Fang et al., 2024; Williams et al., 2025) are promising directions to further improve MLIPs for predicting physical properties.
+
+# 8. Discussion
+
+We identify conservative forces and a smoothly-varying PES as two important properties for MLIPs to consistently perform well in physical property prediction tasks. We offer an analysis of design choices to enhance these two properties. The resulting eSEN architecture bridges the gap between the test-set error and downstream applications, achieving SOTA performance in force/energy prediction, geometry optimization, phonon calculations, and thermal conductivity prediction. This implies it may be possible to use test error as a proxy metric for evaluating model performance during development, if a model passes energy conservation tests. This can accelerate innovations in MLIPs, since benchmarking physical properties usually requires significant domain
+
+knowledge and is usually time-consuming, whereas evaluating test set error is straightforward and efficient.
+
+# Software and Data
+
+Code is available at: https://github.com/facebookresearch/fairchem. Pretrained checkpoints are available at https://huggingface.co/facebook/OMAT24.
+
+# Acknowledgements
+
+We acknowledge Zachary W. Ulissi (FAIR at Meta), Aditi S. Krishnapriyan (UC Berkeley), Samuel M. Blau (LBNL) and other members of the FAIR Chemistry team for helpful discussions, and Ammar Rizvi (FAIR at Meta) for project support.
+
+# Impact Statement
+
+This paper presents work whose goal is to advance the field of machine learning for atomistic modeling, with the potential to accelerate scientific discovery in chemistry and biological sciences. As the field of ML-based atomistic modeling continues to evolve, we emphasize the importance of ongoing discussions on ethical considerations and best practices to ensure that these technologies are developed and applied in ways that benefit society as a whole.
+
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+Zubatyuk, R., Smith, J. S., Leszczynski, J., and Isayev, O. Accurate and transferable multitask prediction of chemical properties with an atoms-in-molecules neural network. Science advances, 5(8):eaav6490, 2019.
+
+# A. Experimental details
+
+# A.1. MD simulation protocol
+
+Simulating molecular systems not seen during training is a key capability of MLIPs. Therefore, we construct challenging MD simulation tasks featuring out-of-distribution data to test a model's conservation capability. We conduct experiments on two chemical domains: inorganic crystals and organic molecules.
+
+Inorganic materials. For training models on inorganic materials we utilize the MPTrj (Deng et al., 2023) dataset, and we establish a suite of simulation tasks based on the TM23 dataset (Owen et al., 2024). TM23 contains MD samples of 27 single vacancy defect transition metal systems at cold, warm, and melt temperatures using an NVT ensemble, in total 81 combinations of different metals and temperature. These defected systems are out-of-distribution for a model trained on the MPTrj datasets, which only contains relaxation trajectories of non-defected systems. Additionally, some of the metals in the TM23 dataset is very rare in the MPTrj dataset. We initialize the simulation by sampling a frame from the TM23 dataset, run a relaxation using the LBFGS algorithm, randomly initialize the atom velocities at cold/warm/melt temperatures using a Maxwell-Boltzmann distribution, then run MD simulations under the NVE ensemble for 100 ps using a time step of 5 fs (same as the time step used in the TM23 ab initio MD protocol).
+
+Organic molecules. For training models on organic molecules we use the SPICE-MACE-OFF dataset (Kovács et al., 2023), which is mainly based on the SPICE-1.0 dataset (Eastman et al., 2023), and we establish a suite of simulation tasks from the MD22 dataset (Chmiela et al., 2023) that contains seven large molecules. Molecules in MD22 are out-of-distribution for a model trained on the SPICE-MACE-OFF dataset as they are considerably larger than all molecules in the SPICE-MACE-OFF training dataset. We initialize the simulation by sampling a frame from the MD22 dataset, run a relaxation using the LBFGS algorithm, randomly initialize the atom velocities at a temperatures of  $400 / 500\mathrm{K}$  using a Maxwell-Boltzmann distribution  $(400\mathrm{K}$  for Buckyball catcher and Double-walled nanotube and  $500\mathrm{K}$  for other molecules, which are the same as the MD22 protocol), then run MD simulations under the NVE ensemble for  $100~\mathrm{ps}$  using a time step of 1 fs (same as the time step used in the MD22 ab initio MD protocol).
+
+All ML-based MD simulations use a Velocity-Verlet integrator and are conducted with ASE (Larsen et al., 2017). We measure the energy conservation error (extent of energy drift) across the 100-ps simulations. All eSEN models are 2-layer with  $L_{\mathrm{max}} = 2$  and  $M_{\mathrm{max}} = 2$  (3.2M trainable parameters). Detailed model hyperparameters are included in Appendix D.
+
+# A.2. Phonon calculation protocols
+
+Harmonic and anharmonic phonon calculations and solutions to the Wigner transport equation (Simoncelli et al., 2022) used for the thermal conductivity benchmark ( $\kappa_{\mathrm{SRME}}$ ) values given in Table 2 were carried out using the supercell method with finite differences implemented in PHONO3PY (Togo et al., 2015; 2023). The calculations followed the protocol described in the Matbench-Discovery benchmark (Riebesell et al., 2023; Póta et al., 2024). For eSEN30M, an even lower  $\kappa_{\mathrm{SRME}}$  of 0.298 is obtained when we adjust the evaluation parameter atom displacement from 0.03 Å to 0.05 Å.
+
+Harmonic phonon calculations for the MDR benchmark results listed in Section 6.2 were carried out following the calculation protocol used by (Loew et al., 2024) which employs phonon calculations using the supercell method with finite differences with a displacement of  $0.01\AA$ . Calculations were done using the PHONOPY software (Togo et al., 2023).
+
+# A.3. Test-set error for MPTrj-trained models
+
+Since MPTrj lacks an official test split and various models are typically trained on distinct subsets of the data, we randomly selected 5000 samples from the subsampled Alexandria (sAlex) dataset (Schmidt et al., 2024; Barroso-Luque et al., 2024) for a fair comparison. This subset was used to calculate the test-set energy mean absolute errors (MAEs) presented in Figures 1, 6, B.7, and B.8.
+
+In the ablation study reported in Table 1, we use a random test split of the MPTrj dataset. We find the performance of eSEN to be highly stable with different random seeds. We trained 2-layer eSENmodels on the MPTrj dataset using 3 different seeds for 50 epochs, with loss coefficients the same loss coefficients of models in Table 1. The validation set errors and standard deviations are  $19.67 \pm 0.23$  for energy (meV/atom),  $43.85 \pm 0.058$  for forces (meV/Å), and  $0.16 \pm 0.00038$  for stress (meV/(Å $^3$ -atom)).
+
+# B. Phonon calculations and vibrational properties
+
+# B.1. Correlation of test-set energy errors and vibrational property errors
+
+Figure B.7 presents the correlation between test-set energy MAE and the other three phonon calculation tasks, evaluated across various model architectures. The corresponding correlations for different variants of eSEN are displayed in Figure B.8. In both figures, improved correlation can be observed among energy-conserving models.
+
+Figure B.8 illustrates that failing the conservation test can
+
+![](images/a1263d55f23b164a08631e7bebdce5ee8304e04cc451c65c79c7134e44451099.jpg)  
+Figure B.7. The correlation between test-set energy error and maximum frequency, free energy and heat capacity across different model architectures.
+
+![](images/728ef8ebe02da3dbf74e2772276dd94c7f95b9de3627a5caf689ab212e744de2.jpg)  
+Figure B.8. The correlation between test-set energy error and maximum frequency, free energy and heat capacity for different variants of eSEN.
+
+result in varying degrees of impact on different properties, depending on the specific design choices made. Although the neighbor limit, envelope function, and number of basis functions substantially affect  $\kappa_{\mathrm{SRME}}$  (see Figure 6), their influence on the properties evaluated in the MDR Phonon benchmark is relatively minor. Representation discretization impacts vibrational entropy and heat capacity but not other properties. While a model might still be able to get good performance in some physical property task when the energy conservation test is failed, when the conservation test is passed, the model performs very well robustly across all metrics.
+
+# B.2. Displacement values and their relation to phonon band structure predictions
+
+Figure B.9 presents the MAEs on phonon calculations for eSEN, MACE, and eqV2 S DeNS as a function of increasing displacement values. As anticipated, both eSEN and MACE exhibit constant or slightly increasing MAE with respect to displacement. In contrast, eqV2 S DeNS displays a notable decrease in MAE with increasing displacement. Notably, when using a displacement of  $0.2\AA$ , the resulting phonon
+
+benchmark MAE values for eqV2 S DeNS become comparable to those of conservative force models (Section 6.2).
+
+While the prediction accuracy of thermodynamic properties such as maximum frequency, entropy, free energy, and heat capacity improves with increasing displacement for direct-force models, this improvement is deceptive and does not translate to accurate predictions of the underlying phonon band structure and density of states (DOS). As illustrated in Figure C.11, the predicted phonon bands and DOS for three selected materials exhibit significant errors, particularly in capturing the correct dispersion relations. Moreover, imaginary frequencies are commonly predicted at small displacement values, suggesting a rough energy landscape (i.e. the learned PES is not truly convex when it's very close to the minima). The eqV2 S DeNS model also fails to accurately capture acoustic modes—those that go to zero linearly at the  $\gamma$  point—which is due to a non-zero net force on the structure. In contrast, non-zero net force at energy local minima does not occur for conservative models by definition (Figure 5).
+
+By enforcing a net zero force prediction, as proposed by Neumann et al. 2024, direct-force models can be modified
+
+![](images/fd951da17d3ab4d74d20b296566db6702681a72b850c3e995b9e392d8709fe1e.jpg)  
+Figure B.9. Errors in a randomly sampled subset (1000 samples) of the MDR Phonon benchmark when the atom displacement is adjusted.
+
+![](images/a3cf144dc53901f6493a4b1477c04454394121bd8a905035a08385a55801e13a.jpg)
+
+![](images/3272bf29b12285c8807d75be914e78346596d7a459bb5c9181108c6093850939.jpg)
+
+![](images/5015255e682a9778893bc090c6a36833fd1e001c7afcd6f3c79d282c1b600fee.jpg)
+
+to accurately capture acoustic phonon modes. As demonstrated in Figure C.12, incorporating this constraint allows the model to predict acoustic modes correctly. However, despite this improvement, the model continues to struggle with accurately reproducing the phonon band structure, and the issue of predicting imaginary frequencies at small displacement values persists.
+
+The apparent paradox of direct-force models like eqV2 S DeNS failing to accurately capture phonon band structures while still achieving competitive accuracy for thermodynamic properties such as entropy, free energy, and heat capacity can be resolved by examining the underlying calculation methodology. These properties are computed using weighted integrals of the DOS. Additionally the metrics in the MDR are performed at room temperature (300 K). The predicted DOS in Figure C.11 and Figure C.12 at larger displacement values adequately captures the overall features of the DFT DOS, but does not reproduce finer details such as high density areas and fluctuations. This level of agreement is sufficient for accurate predictions because the Boltzmann-weighted integrals used in calculating thermal properties help to mitigate the impact of point-wise errors, making it less crucial to precisely capture fine details in the band structure and DOS (Ackland et al., 1997; Van De Walle & Ceder, 2002). Moreover, since properties are estimated at 300 K, models can achieve accurate predictions by prioritizing prediction accuracy of lower-frequency modes, which are more relevant for thermal property calculations, rather than attempting to capture higher-frequency modes. Although eqV2 S DeNS without a net-zero force constraint may not accurately capture acoustic phonon branches, its ability to predict vibrational thermodynamics at room temperature remains unaffected due to the relatively small number of acoustic phonon states at low frequencies.
+
+As a comparison with eqV2 S DeNS, Figures Figure C.13 and Figure C.14 shows the predicted phonon dispersion and DOS for the same three materials using eSEN with direct-force prediction. Although the results still exhibit some of the characteristic artifacts of direct-force models, such
+
+as convergence at larger displacements and the absence of acoustic modes, these issues are less pronounced compared to eqV2 S DeNS. Moreover, the predicted phonon bands and DOS are significantly improved, providing a more accurate representation of the DFT reference values. The better approximation of phonon bands and a lower tendency to predict imaginary frequencies highlight the importance of a smoothly-varying model, even without being conservative.
+
+Extending the existing metrics proposed by Loew et al. 2024 with additional evaluations would provide a more comprehensive assessment of MLIP performance. Specifically, new metrics could be developed to assess phonon dispersion across all modes and frequencies at commensurate points; and computing vibrational thermodynamic properties at a range of temperatures.
+
+![](images/998d8aa8ad76c0a14a6e00fef119fe0f6daccc62915d1c59f9db34f12e9d4206.jpg)  
+Figure B.10. Inference efficiency of MACE-OFF-L and eSENs of a similar scale.
+
+# C. Inference efficiency
+
+We benchmarked the inference speed of our models against the similar sized MACE-OFF-L (Kovács et al., 2023) (4.7M) on a single 80GB Nvidia A-100 GPU. For MACE-OFF-L we used the exact benchmark code found in https://github.com/ACESuit/mace/blob/main/tests/test_benchmark.py with mace-torch v0.3.6 (PyPi). To create a fair comparison, we replicated the identical benchmark environment as MACE benchmarks
+
+using the same diamond system with variable number of supercells (carbon atoms) as input. All models are benchmarked using the standard Python (3.12) runtime with Pytorch v2.4.0 (Paszke et al., 2019) and CUDA 12.1 (Nickolls et al., 2008). No compile/torchscript was used for standardization of runtime. Across all system sizes, eSEN-3.2M has a comparable inference efficiency to MACE-OFF-L. For 216 atoms (Figure B.10), our models (3.2M, 6.4M) can run approximately (0.4, 0.8) million steps per day comparable to MACE-OFF-L (0.7 million steps per day).
+
+# D. Hyper-parameters
+
+Hyper-parameters used for model training are shown in Table 6. We train all models using a per-atom energy MAE loss, a force  $l_{2}$  loss, and a stress MAE loss. For direct-force models or direct-force pre-training, we use the same decomposed loss as described in Barroso-Luque et al. 2024. The eSEN-30M-MP model trained on MPTrj uses Denoising Non-equilibrium Structures (DeNS) (Liao et al., 2024) with a noisig probability of 0.5, a standard deviation of 0.1 Å for the added Gaussian noise, and 10 for the DeNS loss coefficient during direct-force pre-training. DeNS is not used during conservative fine-tuning. In our ablation study for maximum neighbor limit we used 30 as the limit. The main effect of DeNS is reducing overfitting in the pretraining stage. Such benefit is found to carry over to the conservative finetuning stage. The eSEN-30M-OMat and the OAM Finetuned models do not use DeNS.
+
+![](images/3bf5407c5d6fdcfa92f024c6a60929b67f4d387040154cd3ff5f4a0a1be51a56.jpg)  
+Figure C.11. Predicted phonon band structure and density of states (DOS) of Si (diamond structure), CsCl (CsCl structure), AlN (wurtzite structure) using eqV2 S DeNS (direct-force prediction) at different displacement values. DFT baseline is taken from the PBE MDR dataset (Loew et al., 2024) calculated using a displacement of  $0.01\AA$
+
+![](images/7f6cdd88c0bbae250703a9e1d6e9333717bb40529c4a36e2fa68a74694867176.jpg)
+
+![](images/30e495eb83b6f481a71960b75fc37eea93af57313b45f5abf2dc7194dc3cce83.jpg)
+
+![](images/5fcc6defa4a10b2fdb3f477f1739a84f1d77f2355a3ffc09bb31b9be5e059d41.jpg)  
+Figure C.12. Predicted phonon band structure and density of states (DOS) of Si (diamond structure), CsCl (CsCl structure), AlN (wurtzite structure) using eqV2 S DeNS (direct-force prediction with a zero net force constraint) at different displacement values. DFT baseline is taken from the PBE MDR dataset (Loew et al., 2024) calculated using a displacement of  $0.01\AA$
+
+![](images/273524e44416f898ccf33e596a94b977f7fbc99ab92689e43ada8b4aacbb33d5.jpg)
+
+![](images/cf3fe5fc6251fd6bfea4bed605580ddb34727a4b87075c65e6faee3a752ee14a.jpg)
+
+![](images/0ba073d0b0dff7964d5dc99ee9fe55eba59fa3cb47e8e94da48a0a43874a70e3.jpg)  
+Figure C.13. Predicted phonon band structure and density of states (DOS) of Si (diamond structure), CsCl (CsCl structure), AlN (wurtzite structure) using eSEN with direct-force prediction at different displacement values. DFT baseline is taken from the PBE MDR dataset (Loew et al., 2024) calculated using a displacement of  $0.01\AA$
+
+![](images/3456420823f0cc8cdb94f1185c9e8e2e77928e6aea1880178347d950c265cb85.jpg)
+
+![](images/afcfcc27c6293e9ace9d0f7db70dce3ce03836071c77c84e7d3a7a41471132ce.jpg)
+
+![](images/df07e5ee8f4487f0bba7c2cb51e7710c942ec4fc867731f86dbfc4a3da44cbf4.jpg)  
+Figure C.14. Predicted phonon band structure and density of states (DOS) of Si (diamond structure), CsCl (CsCl structure), AlN (wurtzite structure) using eSEN with direct-force prediction and a zero-net force constraint at different displacement values. DFT baseline is taken from the PBE MDR dataset (Loew et al., 2024) calculated using a displacement of  $0.01\AA$
+
+![](images/ee05b20f77532edb7a0fec8a696ee0779929c8ada9cdde30035e3ebfc6a47704.jpg)
+
+![](images/f133dcb9b121d7c6b12260eb9d2e243804b122df481b6412e7d554a18108e95f.jpg)
+
+Table 6. Hyper-parameters for eSEN variants reported in this paper.  ${}^{ * }$  eSEN-30M on MPTrj was trained for 60 epochs using direct-force pre-training and 40 epochs of conserved fine-tuning. eSEN-30M-OMat was trained for 2 epochs using direct-force pre-training and 2 epochs of conserved fine-tuning.  ${}^{ \dagger  }$  The eSEN-30M-OAM model starts from the eSEN-30M-OMat model, and was finetuned for 1 epoch on a dataset constructed by combining the sAlex training dataset and 8 copies of the MPTrj training dataset.  
+
+<table><tr><td>Hyper-parameters</td><td>SPICE-3.2M</td><td>SPICE-6.5M</td><td>MPTrj-3.2M</td><td>MPTrj-6.5M</td><td>MPTrj-30M</td><td>OMat-30M</td><td>OAM Fine-tuning</td></tr><tr><td>Number of eSEN layer blocks</td><td>2</td><td>4</td><td>2</td><td>4</td><td>10</td><td>10</td><td>10</td></tr><tr><td>Maximum degree Lmax</td><td>2</td><td>2</td><td>2</td><td>2</td><td>3</td><td>3</td><td>3</td></tr><tr><td>Maximum order Mmax</td><td>2</td><td>2</td><td>2</td><td>2</td><td>2</td><td>2</td><td>2</td></tr><tr><td>Number of channels Nchannel</td><td>128</td><td>128</td><td>128</td><td>128</td><td>128</td><td>128</td><td>128</td></tr><tr><td>Radial basis function</td><td>Bessel</td><td>Bessel</td><td>Gaussian</td><td>Gaussian</td><td>Gaussian</td><td>Gaussian</td><td>Gaussian</td></tr><tr><td>Number of radial basis functions</td><td>10</td><td>10</td><td>10</td><td>10</td><td>10</td><td>64</td><td>64</td></tr><tr><td>Cutoff radius (Å)</td><td>5</td><td>5</td><td>6</td><td>6</td><td>6</td><td>6</td><td>6</td></tr><tr><td>Batch size</td><td>128</td><td>128</td><td>512</td><td>512</td><td>512</td><td>512</td><td>256</td></tr><tr><td>Optimizer</td><td>AdamW</td><td>AdamW</td><td>AdamW</td><td>AdamW</td><td>AdamW</td><td>AdamW</td><td>AdamW</td></tr><tr><td>Learning rate scheduling</td><td>Cosine</td><td>Cosine</td><td>Cosine</td><td>Cosine</td><td>Cosine</td><td>Cosine</td><td>Cosine</td></tr><tr><td>Warmup epochs</td><td>0.1</td><td>0.01</td><td>0.1</td><td>0.1</td><td>0.1</td><td>0.1</td><td>0.1</td></tr><tr><td>Warmup factor</td><td>0.2</td><td>0.2</td><td>0.2</td><td>0.2</td><td>0.2</td><td>0.2</td><td>0.2</td></tr><tr><td>Maximum learning rate</td><td>4 × 10-4</td><td>4 × 10-4</td><td>4 × 10-4</td><td>4 × 10-4</td><td>4 × 10-4</td><td>4 × 10-4</td><td>2 × 10-4</td></tr><tr><td>Number of epochs</td><td>100</td><td>100</td><td>100</td><td>100</td><td>60 + 40*</td><td>2 + 2*</td><td>1†</td></tr><tr><td>Gradient clipping norm</td><td>100</td><td>100</td><td>100</td><td>100</td><td>100</td><td>100</td><td>100</td></tr><tr><td>Model EMA decay</td><td>0.999</td><td>0.999</td><td>0.999</td><td>0.999</td><td>0.999</td><td>0.999</td><td>0.999</td></tr><tr><td>Weight decay</td><td>1 × 10-3</td><td>1 × 10-3</td><td>1 × 10-3</td><td>1 × 10-3</td><td>1 × 10-3</td><td>1 × 10-3</td><td>1 × 10-3</td></tr><tr><td>Energy loss coefficient</td><td>10</td><td>10</td><td>1</td><td>1</td><td>20</td><td>20</td><td>20</td></tr><tr><td>Force loss coefficient</td><td>20</td><td>20</td><td>10</td><td>10</td><td>20</td><td>20</td><td>20</td></tr><tr><td>Stress loss coefficient</td><td>-</td><td>-</td><td>100</td><td>100</td><td>5</td><td>5</td><td>5</td></tr></table>

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