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NeurIPS/2025/Solving the Asymmetric Traveling Salesman Problem via Trace-Guided Cost Augmentation/full.md

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+# Solving Asymmetric Traveling Salesman Problem via Trace-Guided Cost Augmentation
+
+# Zhen Zhang
+
+$^{1}$ Responsible AI Research Centre Australian Institute for Machine Learning
+
+$^{2}$ The University of Adelaide &  $^{3}$ CSRIO
+
+zhen@zzhang.org
+
+# Javen Qingfeng Shi
+
+$^{1}$ Responsible AI Research Centre Australian Institute for Machine Learning
+
+2The University of Adelaide
+
+javen.shi@adelaide.edu.au
+
+# Wee Sun Lee
+
+School of Computing
+
+National University of Singapore
+
+leews@comp.nus.edu.sg
+
+# Abstract
+
+The Asymmetric Traveling Salesman Problem (ATSP) is one of the most fundamental and notoriously challenging problems in combinatorial optimization. We propose a novel continuous relaxation framework for ATSP that leverages differentiable constraints to encourage acyclic structures and valid permutations. Our approach integrates a differentiable trace-based Directed Acyclic Graph (DAG) constraint with a doubly stochastic matrix relaxation of the assignment problem, enabling gradient-based optimization over soft permutations. We further develop a projected exponentiated-gradient method with adaptive step size to minimize tour cost while satisfying the relaxed constraints. To recover high-quality discrete tours, we introduce a greedy post-processing procedure that iteratively eliminates subtours through cost-aware cycle merging. Empirically, our method achieves state-of-the-art performance on standard asymmetric TSP benchmarks and demonstrates strong scalability and accuracy, particularly on large or highly asymmetric instances where heuristic solvers such as LKH-3 often struggle.
+
+# 1 Introduction
+
+The Asymmetric Traveling Salesman Problem (ATSP) is one of the most fundamental and notoriously challenging problems in combinatorial optimization. Many important combinatorial optimization (CO) problems are theoretically reducible to ATSP. Recent work by Pan et al. [22] has shown that such reductions are not only of theoretical interest but can also be practically feasible for certain small-scale CO tasks, suggesting that advances in ATSP solvers could yield broad benefits across the CO landscape.
+
+Although the symmetric TSP is a special case of ATSP, solvers designed specifically for the asymmetric case often perform poorly on symmetric instances. Conversely, any ATSP instance can be transformed into a symmetric TSP by doubling the number of nodes [9], enabling symmetric solvers to be applied to asymmetric problems. Consequently, much of the CO community has focused on the symmetric TSP [1, 8, 20, 26], leading to the development of increasingly effective handcrafted and learning-based heuristics. Recently, methods such as those proposed by Drakulic et al. [4, 5], Pan et al. [22] have employed reinforcement learning to directly search for optimal ATSP tours. These approaches provide faster inference compared to state-of-the-art heuristics such as LKH-3 [8], but typ
+
+ically yield lower-quality solutions—even though LKH-3 was originally developed for the symmetric TSP.
+
+Assignment-problem (AP) relaxations have long been used to provide strong lower bounds within branch-and-bound and branch-and-cut TSP solvers [17, 19], and they remain effective for large-scale asymmetric instances. Several constructive heuristics, including the Karp-Steele Patching (KSP) algorithm [12, 13] and its variants, attempt to lift assignment solutions into valid tours. However, based on existing open-source implementations, these heuristics often underperform relative to LKH-3. Interestingly, AP relaxations tend to be tighter for asymmetric TSPs than for symmetric ones, motivating the development of stronger heuristics grounded in AP formulations.
+
+A general TSP can be formulated as a linear assignment problem with an additional constraint ensuring that the solution forms exactly one cycle [9]. We observe that this constraint is equivalent to requiring all proper subgraphs of the assignment to be Directed Acyclic Graphs (DAGs). Inspired by recent progress in differentiable DAG learning [25, 28, 29], we propose a novel continuous relaxation of the TSP that incorporates DAG constraints into the assignment formulation. We show that the global minimum of this relaxed problem exactly recovers the true TSP solution.
+
+Although the proposed relaxation is non-convex, we find empirically that local minima obtained via approximate projected gradient descent often yield high-quality solutions. In many cases, our method recovers valid tours with shorter lengths than those produced by LKH-3. For instances where the relaxed optimization does not directly yield a valid tour, we introduce a simple greedy post-processing heuristic to eliminate subtours and construct feasible solutions. Finally, we apply a 2-opt local refinement to further improve tour quality. Together, these components constitute a new heuristic for solving general (asymmetric, non-metric) TSP instances that consistently outperforms LKH-3, particularly on large-scale problems. When applied to small-scale symmetric TSPs, our method remains competitive—achieving slightly lower solution quality than LKH-3 but offering significantly faster runtime performance.
+
+In summary, our main contributions are:
+
+- Novel TSP formulation: A novel exact continuous formulation that suppresses subtours via a trace-based acyclicity loss.  
+- Soft assignment optimization: A doubly stochastic matrix relaxation of ATSP solved via projected exponentiated gradient descent.  
+- Efficient greedy decoding: A post-processing algorithm that reconstructs valid tours from fractional assignments with minimal additional cost.  
+- Strong empirical performance: Extensive experiments show our method outperforms LKH-3 on ATSP benchmarks and scales well to large problem sizes.
+
+# 2 Related Works
+
+Learning-based TSP Solvers Recently, learning-based approaches have been increasingly applied to combinatorial optimization (CO) problems, including the Traveling Salesman Problem (TSP). While many existing works focus on the Euclidean TSP, where inputs are given as point coordinates [11, 14, 15, 18, 20, 23, 24, 31], more recent studies [4, 5, 16, 22] have extended these methods to handle general asymmetric TSPs. Kwon et al. [16] treat distance matrices directly as transformer input tokens, which is computationally inefficient and unsuitable for large graphs. Subsequent methods [4, 5, 22] address this limitation by encoding distance matrices more efficiently, enabling scalability to graphs with thousands of nodes. These approaches [4, 5, 16, 22] generate tours in an auto-regressive manner, formulating the tour-construction process as a Markov Decision Process (MDP). Reinforcement learning is then applied to optimize over the MDP, thereby avoiding reliance on expensive ground-truth optimal solutions.
+
+In contrast to these auto-regressive frameworks, Min et al. [20] proposed a surrogate loss based on assignment-problem relaxations of the TSP. However, their approach is limited to the Euclidean setting and only enforces the absence of length-2 loops in the solution, necessitating a post-processing step to construct valid tours.
+
+Learning-based solvers generally outperform classical heuristics such as LKH-3 in terms of computational efficiency, although they may produce slightly less accurate tours. In reinforcement learning
+
+settings, high-quality solutions from fast heuristics (even if suboptimal) can serve as valuable offline data for off-policy algorithms, potentially accelerating training and improving solution quality.
+
+Beyond methods that directly generate tours from node coordinates or distance matrices, another line of research [27, 30] focuses on learning improved search strategies—such as branch-and-bound or local search policies. These approaches replace handcrafted heuristics in traditional solvers with learned decision mechanisms, leading to more adaptive and data-driven optimization behavior.
+
+Assignment Problem Relaxations for TSP Relaxations of the Traveling Salesman Problem (TSP) based on the classical linear Assignment Problem (AP) have been extensively studied due to their computational efficiency and strong theoretical foundations [9, 17]. These relaxations typically involve solving a linear program in which the tour constraints are relaxed, allowing the solution to contain multiple disconnected subtours. To recover a valid TSP tour, additional strategies such as iterative subtour elimination or branch-and-bound frameworks are employed [19]. Although AP relaxations do not always yield feasible tours directly, they provide tight lower bounds [6] and serve as key components in exact solvers and heuristic algorithms. Several tour-construction heuristics have also been developed based on AP relaxations [12, 13]; however, existing open-source implementations often perform worse than modern LKH-3 solvers. For a more comprehensive discussion, we refer readers to Cirasella et al. [2], Glover et al. [7], and Öncan et al. [21].
+
+# 3 Exact Continuous Relaxation of TSP
+
+Let  $\mathbf{D} \in \mathbb{R}_+^{n \times n}$  be the non-negative distance (or cost) matrix, where its entry  $d_{ij}$  denotes the distance or cost of traveling from node  $i$  to node  $j$ . The general TSP seeks a tour (i.e., a permutation of the nodes forming a single cycle that visits each node exactly once) that minimizes the total travel cost. This can be formulated as:
+
+$$
+\begin{array}{l} \min  _ {\mathbf {X} \in \{0, 1 \} ^ {n \times n}} \quad \langle \mathbf {D}, \mathbf {X} \rangle = \sum_ {i = 1} ^ {n} \sum_ {j = 1} ^ {n} d _ {i j} x _ {i j} \\ \text {s u b j e c t} \quad \sum_ {j = 1} ^ {n} x _ {i j} = 1 \quad \text {f o r a l l} i \in \{1, \dots , n \} \quad (\text {e a c h n o d e e x i t s e x a c t l y o n c e}), \tag {1} \\ \sum_ {i = 1} ^ {n} x _ {i j} = 1 \quad \text {f o r a l l} j \in \{1, \dots , n \} \quad (\text {e a c h n o d e i s e n t e r e d e x a c t l y o n c e}), \\ \mathbf {X} \text {f o r m s a s i n g l e H a m i t o n i a n c y c l e (n o s u b t o u r s)}. \\ \end{array}
+$$
+
+Here,  $\mathbf{X}$  is a permutation matrix corresponding to a single cycle over the nodes. The final constraint prohibits subtours—i.e., disjoint cycles that do not span all nodes—ensuring that the solution corresponds to a valid TSP tour.
+
+Differentiable DAG Constraints Recently, a family of differentiable constraints for enforcing acyclicity in graphs was proposed in the literature [25, 28, 29]. For any positive adjacency matrix  $\mathbf{X}$ , acyclicity can be enforced by the following constraint:
+
+$$
+h (\mathbf {X}) = \sum_ {i = 1} ^ {n} \operatorname {t r} \left(\mathbf {X} ^ {i}\right) = 0. \tag {2}
+$$
+
+In this constraint, each term  $\mathrm{tr}(\mathbf{X}^i)$  is strictly positive when  $\mathbf{X}$  contains cycles of length  $i$ , since all entries of  $\mathbf{X}$  are positive. Therefore, setting the total sum to zero guarantees that there are no cycles of any length, ensuring that  $\mathbf{X}$  represents a Directed Acyclic Graph (DAG).
+
+Based on these differentiable DAG constraints, we propose a novel continuous relaxation of the TSP by enforcing that any proper subgraph of the assignment matrix  $\mathbf{X}$  forms a DAG. This leads to the following formulation:
+
+$$
+\min  _ {\mathbf {X} \in \mathcal {D} ^ {n}} \left\langle \mathbf {D}, \mathbf {X} \right\rangle = \sum_ {i = 1} ^ {n} \sum_ {j = 1} ^ {n} d _ {i j} x _ {i j} \quad \text {s u b j e c t t o} \quad \sum_ {i = 1} ^ {n - 1} \operatorname {t r} \left(\mathbf {X} ^ {i}\right) = 0, \tag {3}
+$$
+
+where  $\mathcal{D}^n = \left\{\mathbf{X}\in [0,1]^{n\times n}\mid \mathbf{X}\mathbf{1} = \mathbf{1},\mathbf{1}^\top \mathbf{X} = \mathbf{1}^\top \right\}$  is the set of doubly stochastic matrices of size  $n\times n$
+
+One appealing property of this relaxation is its tightness, as stated below:
+
+Proposition 1 (Exactness of the Continuous Relaxation). The global minimum of the relaxed problem in (3) coincides with the optimal solution of the original TSP.
+
+Proof. To prove the result, it suffices to show that any fractional matrix  $\mathbf{X} \in \mathcal{D}^n$  cannot satisfy the constraint  $\sum_{i=1}^{n-1} \operatorname{tr}(\mathbf{X}^i) = 0$ . In other words, all feasible solutions must be permutation matrices representing Hamiltonian cycles.
+
+The proof follows from the Birkhoff-von Neumann theorem [10], which states that any doubly stochastic matrix  $\mathbf{X} \in \mathcal{D}^n$  can be expressed as a convex combination of permutation matrices:
+
+$$
+\mathbf {X} = \sum_ {i = 1} ^ {k} \theta_ {i} \mathbf {P} _ {i}, \quad \theta_ {i} \geq 0, \sum_ {i = 1} ^ {k} \theta_ {i} = 1, \tag {4}
+$$
+
+where  $\mathbf{P}_i$  are permutation matrices.
+
+Assume for contradiction that a fractional  $\mathbf{X} \in \mathcal{D}^n$  satisfies the constraint in (3). Then its Birkhoff-von Neumann decomposition contains at least two distinct permutation matrices  $\mathbf{P}_i$ . If any  $\mathbf{P}_i$  corresponds to a subtour (i.e., not a single Hamiltonian cycle), then  $\mathrm{tr}(\mathbf{P}_i^j) > 0$  for some  $j < n$ , and hence  $\mathrm{tr}(\mathbf{X}^j) > 0$  for some  $j < n$  due to linearity and positivity. This violates the constraint  $\sum_{j=1}^{n-1} \mathrm{tr}(\mathbf{X}^j) = 0$ , making  $\mathbf{X}$  infeasible.
+
+Even if all  $\mathbf{P}_i$  represent valid Hamiltonian cycles, a convex combination of two or more distinct tours will produce fractional entries in  $\mathbf{X}$ . These fractional values imply the presence of edges from multiple tours, which results the presence of subtours and nonzero traces in  $\mathbf{X}^j$  for some  $j < n$ . Thus,  $\mathbf{X}$  would again be infeasible.
+
+Therefore, the only feasible solutions are permutation matrices representing single Hamiltonian cycles. Among them, the one minimizing  $\langle \mathbf{D},\mathbf{X}\rangle$  is exactly the TSP solution.
+
+![](images/20d57ad8c85ab32957a217096c85f86252b68a329fe174be939cd0a772443bb4.jpg)
+
+# 4 Trace Guided Tour Search
+
+To efficiently optimize the relaxation in Equation (3), we introduce an auxiliary variable  $\mathbf{Y} \in \mathbb{R}^{n \times n}$  with elementwise constraints  $\mathbf{Y} \leqslant 0$ , and define the assignment matrix as  $\mathbf{X} = \exp(\mathbf{Y})$ , where the exponential is applied elementwise. This reparameterization ensures that  $\mathbf{X}$  is strictly positive and facilitates optimization over the interior of the doubly stochastic polytope  $\mathcal{D}^n$ . Compared to optimizing directly over  $\mathbf{X}$ , working in the log-domain (i.e., optimizing  $\mathbf{Y}$ ) avoids boundary issues, improves numerical stability, and enables smooth gradients that are more amenable to gradient-based learning methods. The relaxed optimization problem is then formulated as:
+
+$$
+\min  _ {\exp (Y) \in \mathcal {D} ^ {n}} \langle \mathbf {D}, \exp (\mathbf {Y}) \rangle = \sum_ {i = 1} ^ {n} \sum_ {j = 1} ^ {n} d _ {i j} \exp \left(y _ {i j}\right) \text {s u b j e c t t o} \operatorname {t r} \sum_ {i = 1} ^ {n - 1} \left(\exp (\mathbf {Y}) ^ {i}\right) = 0, \tag {5}
+$$
+
+where  $\mathbf{D}$  is the cost matrix, and the trace constraint eliminates self-loops and short cycles. By introducing a Lagrange multiplier  $\lambda$ , the constrained problem can be rewritten as the following unconstrained objective:
+
+$$
+\min  _ {\exp (\mathbf {Y}) \in \mathcal {D} ^ {n}} f (\mathbf {Y}) = \langle \mathbf {D}, \exp (\mathbf {Y}) \rangle + \lambda \operatorname {t r} \sum_ {i = 1} ^ {n - 1} \left(\exp (\mathbf {Y}) ^ {i}\right). \tag {6}
+$$
+
+The gradient of this objective with respect to  $\mathbf{Y}$  is given by:
+
+$$
+\nabla_ {\mathbf {Y}} f (\mathbf {Y}) = \mathbf {D} \odot \exp (\mathbf {Y}) + \lambda \left[ \sum_ {i = 0} ^ {n - 2} (i + 1) \exp (\mathbf {Y}) ^ {i} \right] ^ {\top} \odot \exp (\mathbf {Y}), \tag {7}
+$$
+
+where  $\odot$  denotes elementwise multiplication. This formulation enables efficient and scalable optimization while preserving the structural constraints required by the TSP.
+
+Algorithm 1 Approximate Projected Gradient Descent for Exponentiated TSP Relaxation  
+1: Input: Cost matrix  $\mathbf{D} \in \mathbb{R}^{n \times n}$ , initial step size  $\alpha > 0$ , scaling factor  $\gamma > 0$ , max iterations  $T$   
+2: Initialize  $\mathbf{Y} \gets -\mathbf{D}$   
+3: Compute initial assignment  $\pi_{\mathrm{prev}} \gets \mathrm{LinearAssignment}(-\mathbf{Y})$   
+4: for  $t = 1$  to  $T$  do  
+5: Compute gradient:  
+ $\nabla_{\mathbf{Y}} f(\mathbf{Y}) = \mathbf{D} \odot \exp(\mathbf{Y}) + \lambda \left[ \sum_{i=0}^{n-2} (i+1) \exp(\mathbf{Y})^i \right]^\top \odot \exp(\mathbf{Y})$   
+6: Update  $\mathbf{Y}_{\mathrm{new}} \gets \mathbf{Y} - \alpha \nabla_{\mathbf{Y}} f(\mathbf{Y})$   
+7: Project:  $\mathbf{X} = \mathrm{Sinkhorn}(\exp(\mathbf{Y}_{\mathrm{new}}))$ ,  $\mathbf{Y}_{\mathrm{new}} \gets \log(\mathbf{X}) \triangleright$  elementwise projection onto  $\mathbf{D}^n$   
+8: Compute new assignment  $\pi_{\mathrm{new}} \gets \mathrm{LinearAssignment}(-\mathbf{Y}_{\mathrm{new}})$   
+9: if  $\pi_{\mathrm{new}} \neq \pi_{\mathrm{prev}}$  then  
+10:  $\alpha \gets \alpha / 2$   
+11: else  
+12:  $\alpha \gets 2\alpha$   
+13: end if  
+14:  $\mathbf{Y} \gets \mathbf{Y}_{\mathrm{new}}, \pi_{\mathrm{prev}} \gets \pi_{\mathrm{new}}$   
+15: if  $\pi_{\mathrm{new}}$  has only one loop then  
+16: break  
+17: end if  
+18: end for  
+19: Output:  $\mathbf{Y}$ , permutation  $\pi_{\mathrm{new}}$
+
+Approximate Projected Gradient Descent for Exponentiated TSP Relaxation: To optimize the exponentiated relaxation of the TSP problem, we propose an approximate projected gradient descent algorithm over the log-domain variable  $\mathbf{Y}$ . The algorithm initializes  $\mathbf{Y}$  as the negative of the cost matrix,  $\mathbf{Y} \gets -\mathbf{D}$ , and iteratively updates it using the gradient of the relaxed objective. At each step, the gradient is computed and used to update  $\mathbf{Y}$  with a step size  $\alpha$ . To maintain feasibility within the doubly stochastic set, we apply Sinkhorn [3] normalization to the exponentiated matrix  $\exp(\mathbf{Y})$  and then project back to the log-domain via  $\log(\cdot)$ , ensuring that  $\exp(\mathbf{Y}) \in \mathcal{D}^n$ . The step size  $\alpha$  is dynamically adjusted based on whether the solution to the linear assignment problem (using  $-\mathbf{Y}$  as the cost) changes after the update: if it does, the step size is reduced; otherwise, it is increased to accelerate convergence. This adaptive strategy helps balance exploration and stability. The process terminates either after a fixed number of iterations or once a valid TSP tour (a single loop) is found in the assignment. The algorithm returns the optimized  $\mathbf{Y}$  and the corresponding assignment matrix  $\exp(\mathbf{Y})$ .
+
+Approximate Gradient Computation: In our gradient expression, the term  $\left[\sum_{i=0}^{n-2}(i+1)\exp(\mathbf{Y})^i\right]^\top$  involves computing a weighted sum of matrix powers, which can be computationally expensive. Although efficient approximations such as the truncated Neumann series proposed in Zhang et al. [28] can reduce the number of operations to  $O(\log n)$  matrix multiplications, the cost remains significant for large instances due to the  $O(n^3)$  complexity of matrix multiplication. To address this, we adopt a lightweight approximation by substituting  $\exp(\mathbf{Y})$  with the hard assignment matrix corresponding to the current solution  $\pi_{\mathrm{prev}}$ . Since permutation matrices are sparse and binary, their powers are simply cyclic shifts, and the sum can be computed in  $O(n^2)$  time. This approximation significantly accelerates training while preserving useful structural signals from the current assignment during optimization.
+
+Greedy Tour Decoding from Invalid Assignments: Although the proposed optimization approach typically converges rapidly for asymmetric TSP instances, there exist hard cases where convergence to a valid permutation (i.e., a single-loop tour) does not occur. To address such situations, we apply a greedy decoding strategy to extract a valid TSP tour from the relaxed assignment matrix. Specifically, we begin by selecting a random starting node and iteratively follow the highest-assigned edge (i.e., the node with the largest assignment probability) to determine the next node in the tour. If the selected
+
+<table><tr><td rowspan="2">Method</td><td colspan="3">ATSP20</td><td colspan="3">ATSP50</td></tr><tr><td>Avg. GAP</td><td>Median GAP</td><td>Runtime (s)</td><td>Avg. GAP</td><td>Median GAP</td><td>Runtime (s)</td></tr><tr><td>LKH-3 (10, 10000)</td><td>0.00%</td><td>0.00%</td><td>1.68</td><td>0.00%</td><td>0.00%</td><td>5.18</td></tr><tr><td>LKH-3 (1, 500)</td><td>0.02%</td><td>0.00%</td><td>0.02</td><td>1.00%</td><td>0.40%</td><td>0.060</td></tr><tr><td>LKH-3 (1, 10000)</td><td>0.0005%</td><td>0.00%</td><td>0.18</td><td>0.28%</td><td>0.00%</td><td>0.54</td></tr><tr><td>MatPOENet</td><td>27.36%</td><td>27.80%</td><td>0.0067</td><td>688.67%</td><td>675.04%</td><td>0.016</td></tr><tr><td>GOAL</td><td>36.50%</td><td>33.59%</td><td>0.002</td><td>108.29%</td><td>109.38%</td><td>0.0095</td></tr><tr><td>Ours</td><td>1.9%</td><td>0.08%</td><td>0.005</td><td>2.6%</td><td>0.8%</td><td>0.012</td></tr></table>
+
+Table 1: Comparison of methods on small-scale pure random ATSP (TSP20 and TSP50). GAP is measured relative to LKH-3 (10, 10000). The results are average from 128 instances. The performance of learning based approaches is far worse than their reported results as they are trained on ATSP with triangle inequality and they failed to generalize to random ATSP.  
+
+<table><tr><td rowspan="2">Method</td><td colspan="3">ATSP20</td><td colspan="3">ATSP50</td></tr><tr><td>Avg. GAP</td><td>Median GAP</td><td>Runtime (s)</td><td>Avg. GAP</td><td>Median GAP</td><td>Runtime (s)</td></tr><tr><td>LKH-3 (10, 10000)</td><td>0.00%</td><td>0.00%</td><td>2.54</td><td>0.00%</td><td>0.00%</td><td>4.51</td></tr><tr><td>LKH-3 (1, 500)</td><td>0.00%</td><td>0.00%</td><td>0.03</td><td>0.06%</td><td>0.00%</td><td>0.05</td></tr><tr><td>LKH-3 (1, 10000)</td><td>0.00%</td><td>0.00%</td><td>0.27</td><td>0.004%</td><td>0.00%</td><td>0.48</td></tr><tr><td>MatPOENet</td><td>7.4%</td><td>5.3%</td><td>0.0063</td><td>1.4%</td><td>1.1%</td><td>0.016</td></tr><tr><td>GOAL</td><td>9.26%</td><td>10.15%</td><td>0.002</td><td>5.87%</td><td>5.83%</td><td>0.0095</td></tr><tr><td>Ours</td><td>0.97%</td><td>0.00%</td><td>0.006</td><td>0.6%</td><td>0.00%</td><td>0.009</td></tr></table>
+
+Table 2: Comparison of methods on small-scale random ATSP with triangle inequality problems (TSP20 and TSP50). GAP is measured relative to LKH-3 (10, 10000). The results are average from 128 instances.
+
+edge would result in the formation of a premature loop or revisiting a node, we instead choose the closest unvisited node according to the original cost matrix. This process continues until all nodes have been visited exactly once, ensuring the construction of a valid Hamiltonian cycle. This decoding heuristic is simple, efficient, and empirically effective in recovering feasible tours from imperfect or fractional assignments.
+
+Time Complexity. The dominant computational cost in each iteration of the proposed algorithm lies in solving the linear assignment problem for approximating  $\exp (\mathbf{Y})$ , which has a worst-case time complexity of  $O(n^{3})$  when using the Hungarian algorithm. However, in practice, the Hungarian algorithm is typically much faster due to early termination and efficient implementations. Additionally, viewing the assignment step through the lens of primal-dual linear programming reveals an opportunity to reuse the dual variables as warm starts for subsequent iterations, potentially accelerating convergence. Alternatively, the assignment problem can also be solved using the Auction algorithm, which offers better parallelism and is more amenable to GPU implementation. This opens up promising directions for future work, where the entire optimization process could be ported to GPUs to achieve significant speed-ups and scalability for large-scale TSP instances.
+
+Local Search Refinement. To further improve the quality of the obtained tour, our implementation applies a simple 2-opt heuristic as a post-processing step. We observe that for asymmetric TSP instances, the 2-opt heuristic often brings negligible improvements due to the directional nature of the cost structure. In contrast, for symmetric TSPs, the 2-opt refinement can lead to noticeable performance gains by eliminating crossing edges and reducing the overall tour length. While 2-opt is lightweight and easy to implement, it may be suboptimal for larger or more complex instances. As a future direction, one could consider replacing the 2-opt heuristic with more powerful local search methods such as the Lin-Kernighan-Helsgaun (LKH) algorithm [8]. In such a setup, our learned solution could serve as a high-quality initialization for LKH, potentially reducing the number of local search iterations required and improving overall efficiency.
+
+# 5 Experimental Results
+
+Hardware All experiments were conducted on a workstation equipped with an AMD Ryzen 9 3900X CPU, 128GB DDR4 memory, and an NVIDIA RTX 3090 GPU. The LKH-3 heuristic was implemented in  $\mathrm{C + + }$  and executed in a single-threaded mode on the CPU. Our proposed method was implemented using Python and SciPy, and it was also executed on a single CPU process without utilizing the GPU. In contrast, all learning-based approaches (e.g., GOAL and MatPOENet) were executed on the GPU to leverage hardware acceleration during inference.
+
+Dataset We evaluate our method on randomly generated Asymmetric TSP (ATSP) instances of sizes 20, 50, 100, 200, 500, and 1000. Following prior works [16], the cost matrices for these instances are generated by sampling each entry independently from a uniform distribution over the interval  $[0,1]$ . Kwon et al. [16] add an post-processing procedure to ensure the triangle inequality in enforced in the generated ATSP, while we also consider the cases where the triangle inequality is not satisfied. For solvers that require integer-valued cost matrices, we scale the costs by  $10^{6}$  and round them to the nearest integer to preserve precision. Additionally, we evaluate our approach on symmetric TSP instances of sizes 20 and 50. In line with previous settings, we first sample  $n$  random points in 2D space from a uniform distribution over  $[0,1]^{2}$ , and use the Euclidean distances between these points to construct the symmetric cost matrices.
+
+Competitors We compare our method against several strong baselines, including both classical heuristics and recent learning-based approaches. Specifically, we consider LKH-3, a state-of-the-art heuristic algorithm for solving TSP, GOAL[5], a reinforcement learning-based approach using graph neural networks, and MatPOENet[22], a transformer-based model for solving combinatorial optimization problems. For GOAL and MatPOENet, we use their publicly released pretrained weights to ensure a fair comparison. For LKH-3, we evaluate three configurations: (i) one run with 500 search trials, (ii) one run with 10,000 search trials, and (iii) ten runs with 10,000 search trials each. These configurations allow us to assess both the performance and robustness of LKH-3 under varying computational budgets. Exact solvers such as Gurobi and Concorde [1] are not compared as they are often too slow for large scale problems.
+
+<table><tr><td rowspan="2">Method</td><td colspan="3">ATSP100</td><td colspan="3">ATSP200</td></tr><tr><td>Avg. GAP</td><td>Median GAP</td><td>Runtime (s)</td><td>Avg. GAP</td><td>Median GAP</td><td>Runtime (s)</td></tr><tr><td>LKH-3 (10, 10000)</td><td>0.00%</td><td>0.00%</td><td>11.40</td><td>0.00%</td><td>0.00%</td><td>18.67</td></tr><tr><td>LKH-3 (1, 500)</td><td>5.38%</td><td>5.06%</td><td>0.13</td><td>12.59%</td><td>12.72%</td><td>0.27</td></tr><tr><td>LKH-3 (1, 10000)</td><td>1.06%</td><td>0.91%</td><td>1.19</td><td>2.02%</td><td>1.90%</td><td>2.03</td></tr><tr><td>MatPOENet</td><td>295.18%</td><td>295.62%</td><td>0.047</td><td>747.56%</td><td>750.27%</td><td>0.165</td></tr><tr><td>GOAL</td><td>181.92%</td><td>177.47%</td><td>0.058</td><td>265.03%</td><td>261.43%</td><td>0.16</td></tr><tr><td>Ours</td><td>3.39%</td><td>1.77%</td><td>0.043</td><td>0.32%</td><td>-1.01%</td><td>0.13</td></tr></table>
+
+Table 3: Comparison of methods on medium-scale pure random ATSP problems (TSP100 and TSP200). GAP is measured relative to LKH-3 (10, 10000). The results are average from 128 instances. The performance of learning based approaches is far worse than their reported results as they are trained on ATSP with triangle inequality and they failed to generalize to random ATSP.
+
+# 5.1 Asymmetric Problems
+
+Small Scale Problems The comparison of running time and solution quality for small-scale problems is presented in Table 1 and Table 2. Our method, while yielding slightly worse average performance than LKH-3, demonstrates significantly faster running times. For the learning-based approaches, which are trained on ATSP with triangle inequality instances, their performance degrades substantially on purely random ATSP problems—underperforming all other competitors. This indicates that current neural combinatorial solvers may struggle to generalize across different problem distributions. Additionally, MatPOENet performs considerably worse on our test set than reported in its original paper, suggesting a potential distribution shift between the two test scenarios.
+
+Surprisingly, GOAL achieves the best performance on asymmetric TSP20 with triangle inequality, outperforming the state-of-the-art LKH-3 heuristic by a substantial margin. Given that GOAL was
+
+<table><tr><td rowspan="2">Method</td><td colspan="3">ATSP100</td><td colspan="3">ATSP200</td></tr><tr><td>Avg. GAP</td><td>Median GAP</td><td>Runtime (s)</td><td>Avg. GAP</td><td>Median GAP</td><td>Runtime (s)</td></tr><tr><td>LKH-3 (10, 10000)</td><td>0.00%</td><td>0.00%</td><td>7.31</td><td>0.00%</td><td>0.00%</td><td>11.69</td></tr><tr><td>LKH-3 (1, 500)</td><td>0.25%</td><td>0.11%</td><td>0.11</td><td>0.64%</td><td>0.06%</td><td>0.23</td></tr><tr><td>LKH-3 (1, 10000)</td><td>0.09%</td><td>0.00%</td><td>0.787</td><td>0.18%</td><td>0.01%</td><td>1.34</td></tr><tr><td>MatPOENet</td><td>14.91%</td><td>13.84%</td><td>0.047</td><td>35.58%</td><td>35.72%</td><td>0.165</td></tr><tr><td>GOAL</td><td>3.83%</td><td>4.04%</td><td>0.058</td><td>0.77%</td><td>0.74%</td><td>0.44</td></tr><tr><td>Ours</td><td>0.52%</td><td>0.04%</td><td>0.027</td><td>0.16%</td><td>0.00%</td><td>0.10</td></tr></table>
+
+Table 4: Comparison of methods on medium-scale random ATSP with triangle inequality problems (TSP100 and TSP200). GAP is measured relative to LKH-3 (10, 10000). The results are average from 128 instances. There may potentially be a distribution shift between the training set of MatPOENet and our test set.  
+
+<table><tr><td rowspan="2">Method</td><td colspan="3">ATSP500</td><td colspan="3">ATSP1000</td></tr><tr><td>Avg. GAP</td><td>Median GAP</td><td>Runtime (s)</td><td>Avg. GAP</td><td>Median GAP</td><td>Runtime (s)</td></tr><tr><td>LKH-3 (10, 10000)</td><td>0.00%</td><td>0.00%</td><td>29.99</td><td>0.00%</td><td>0.00%</td><td>46.64</td></tr><tr><td>LKH-3 (1, 500)</td><td>24.47%</td><td>24.86%</td><td>1.08</td><td>51.23%</td><td>51.48%</td><td>3.81</td></tr><tr><td>LKH-3 (1, 10000)</td><td>2.50%</td><td>2.45%</td><td>3.85</td><td>3.25%</td><td>2.94%</td><td>7.91</td></tr><tr><td>MatPOENet</td><td>6400.51%</td><td>6400.50%</td><td>1.51</td><td>15017.16%</td><td>15016.70%</td><td>10.24</td></tr><tr><td>GOAL</td><td>369.69%</td><td>369.40%</td><td>6.84</td><td>305.84%</td><td>304.62%</td><td>58.53</td></tr><tr><td>Ours</td><td>-10.77 %</td><td>-11.59%</td><td>0.87</td><td>-21.40%</td><td>-21.86%</td><td>5.22</td></tr></table>
+
+Table 5: Comparison of methods on large-scale pure random ATSP problems (TSP500 and TSP1000). GAP is measured relative to LKH-3 (10, 10000). The results are average from 128 instances. Our approaches outperforms the state-of-the-arts LKH-3 heuristics in terms of solution quality. The performance of learning based approaches is far worse than their reported results as they are trained on ATSP with triangle inequality and they failed to generalize to random ATSP.
+
+trained on 100-node triangle asymmetric instances, this result suggests that training neural solvers on large, structured, and challenging problems may enhance their generalization to smaller problem instances.
+
+Medium and Large Scale Problems The comparison of running time and solution quality for medium- and large-scale problems is presented in Table 3, Table 4, Table 5, and Table 6. Consistent with the small-scale setting, learning-based approaches perform significantly worse than others on purely random asymmetric TSPs. In these settings, our algorithm outperforms LKH-3(1,500) for graphs with more than 100 nodes, and even surpasses LKH-3(1,10000) for graphs with more than 200 nodes. As the graph size increases, our method increasingly outperforms all LKH-3 heuristics, with the performance gap becoming more pronounced for larger instances. On 1000-node problems, our algorithm achieves over  $20\%$  improvement compared to LKH-3(10,10000) and over  $70\%$  compared to LKH-3(1,500). Meanwhile, our runtime on large graphs is comparable to that of LKH-3(1,500) and remains faster than the learning-based approaches. On asymmetric TSPs with triangle inequality, similar trends are observed. Our method continues to outperform LKH-3 on large graphs, although the performance margin is smaller compared to the purely random setting.
+
+In terms of running time, it is notable that, despite our implementation not being carefully optimized, its execution speed is often faster than that of LKH-3. On very large graphs, our method can outperform learning-based approaches, even when those methods are accelerated by GPUs. Furthermore, most operations in our algorithm are amenable to parallel execution and could potentially benefit significantly from GPU acceleration.
+
+# 5.2 Symmetric Problems
+
+Although our algorithm is not specifically designed for symmetric TSPs and does not exploit any symmetry properties, we evaluated its performance on small-scale symmetric instances. The results are presented in Table 7. The performance trends closely resemble those observed on triangle asymmetric TSPs.
+
+<table><tr><td rowspan="2">Method</td><td colspan="3">ATSP500</td><td colspan="3">ATSP1000</td></tr><tr><td>Avg. GAP</td><td>Median GAP</td><td>Runtime (s)</td><td>Avg. GAP</td><td>Median GAP</td><td>Runtime (s)</td></tr><tr><td>LKH-3 (10, 10000)</td><td>0.00%</td><td>0.00%</td><td>20.88</td><td>0.00%</td><td>0.00%</td><td>36.72</td></tr><tr><td>LKH-3 (1, 500)</td><td>1.08%</td><td>1.00%</td><td>1.02</td><td>51.23%</td><td>51.48%</td><td>3.93</td></tr><tr><td>LKH-3 (1, 10000)</td><td>0.25%</td><td>0.24%</td><td>2.93</td><td>3.25%</td><td>2.94%</td><td>7.05</td></tr><tr><td>MatPOENet</td><td>112.48%</td><td>112.19%</td><td>1.50</td><td>156.48%</td><td>156.24%</td><td>10.21</td></tr><tr><td>GOAL</td><td>1.07%</td><td>1.01%</td><td>6.85</td><td>1.26%</td><td>1.22%</td><td>58.48</td></tr><tr><td>Ours</td><td>-0.06 %</td><td>-0.11%</td><td>0.74</td><td>-0.35%</td><td>-0.36%</td><td>5.10</td></tr></table>
+
+Table 6: Comparison of methods on large-scale random ATSP with triangle inequality problems (TSP500 and TSP1000). GAP is measured relative to LKH-3 (10, 10000). The results are average from 128 instances. Our approaches outperforms the state-of-the-arts LKH-3 heuristics in terms of solution quality.
+
+<table><tr><td rowspan="2">Method</td><td colspan="3">TSP20</td><td colspan="3">TSP50</td></tr><tr><td>Avg. GAP</td><td>Median GAP</td><td>Runtime (s)</td><td>Avg. GAP</td><td>Median GAP</td><td>Runtime (s)</td></tr><tr><td>LKH-3 (10, 10000)</td><td>0.00%</td><td>0.00%</td><td>3.21</td><td>0.00%</td><td>0.00%</td><td>4.51</td></tr><tr><td>LKH-3 (1, 500)</td><td>0.00%</td><td>0.00%</td><td>0.03</td><td>0.06%</td><td>0.00%</td><td>0.05</td></tr><tr><td>LKH-3 (1, 10000)</td><td>0.00%</td><td>0.00%</td><td>0.33</td><td>0.004%</td><td>0.00%</td><td>0.48</td></tr><tr><td>MatPOENet</td><td>1.50%</td><td>1.25%</td><td>0.0064</td><td>0.63%</td><td>0.58%</td><td>0.015</td></tr><tr><td>GOAL</td><td>14.06%</td><td>15.14%</td><td>0.002</td><td>14.34%</td><td>15.18%</td><td>0.0095</td></tr><tr><td>Ours</td><td>0.04%</td><td>0.00%</td><td>0.013</td><td>0.86%</td><td>0.66%</td><td>0.094</td></tr></table>
+
+Table 7: Comparison of methods on small-scale random symmetric TSP problems (TSP20 and TSP50). GAP is measured relative to LKH-3 (10, 10000). The results are average from 128 instances.
+
+# 6 Conclusions and Future Works
+
+In this work, we introduced a novel continuous relaxation framework for the Asymmetric Traveling Salesman Problem (ATSP) that combines differentiable acyclicity constraints with doubly stochastic matrix relaxations. Our key innovation lies in the formulation of ATSP as a constrained optimization problem where trace-based DAG constraints naturally eliminate subtours while maintaining differentiability for gradient-based optimization. The proposed projected exponentiated gradient method with adaptive step sizes demonstrates efficient convergence properties, and our greedy post-processing procedure effectively recovers valid tours from fractional assignments.
+
+Through extensive experiments on standard benchmarks, we have shown that our method achieves state-of-the-art performance on asymmetric TSP instances, particularly excelling on large-scale problems where traditional heuristics like LKH-3 struggle. The results reveal several important insights: (1) our continuous relaxation provides a tight formulation that frequently converges to valid tours without post-processing, (2) the approach scales favorably with problem size, maintaining both computational efficiency and solution quality, and (3) the method remains competitive even on symmetric TSP instances despite not being specifically designed for them.
+
+The success of our framework suggests several promising directions for future research. First, the differentiable nature of our formulation makes it potentially amenable to integration with learning-based approaches, where the solver could be trained to predict good initializations or adaptive penalty parameters. Second, the trace-based constraints may generalize to other combinatorial optimization problems that require acyclic substructures. Finally, investigating more sophisticated optimization techniques within our framework could lead to further improvements in both solution quality and computational efficiency.
+
+This work bridges an important gap between continuous relaxations and discrete combinatorial optimization, offering a new perspective on solving challenging problems like ATSP through differentiable programming. The demonstrated effectiveness of our approach on large-scale instances suggests that similar techniques could prove valuable for other NP-hard optimization problems where traditional methods face scalability limitations.
+
+# Acknowledgments and Disclosure of Funding
+
+This research is supported by Responsible AI Research Center (RAIR) and the Ministry of Education, Singapore, under its Academic Research Fund Tier 1 (A-8001814-00-00).
+
+# References
+
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+[24] Zhiqing Sun and Yiming Yang. Difusco: Graph-based diffusion solvers for combinatorial optimization. Advances in neural information processing systems, 36:3706-3731, 2023.
+
+[25] Dennis Wei, Tian Gao, and Yue Yu. Dags with No Fears: A closer look at continuous optimization for learning bayesian networks. Advances in Neural Information Processing Systems, 33:3895-3906, 2020.  
+[26] Yifan Xia, Xianliang Yang, Zichuan Liu, Zhihao Liu, Lei Song, and Jiang Bian. Position: Rethinking post-hoc search-based neural approaches for solving large-scale traveling salesman problems. arXiv preprint arXiv:2406.03503, 2024.  
+[27] Haoran Ye, Jiarui Wang, Helan Liang, Zhiguang Cao, Yong Li, and Fanzhang Li. Glop: Learning global partition and local construction for solving large-scale routing problems in real-time. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 38, pages 20284-20292, 2024.  
+[28] Zhen Zhang, Ignavier Ng, Dong Gong, Yuhang Liu, Ehsan Abbasnejad, Mingming Gong, Kun Zhang, and Javen Qinfeng Shi. Truncated matrix power iteration for differentiable dag learning. Advances in Neural Information Processing Systems, 35:18390-18402, 2022.  
+[29] Zhen Zhang, Ignavier Ng, Dong Gong, Yuhang Liu, Mingming Gong, Biwei Huang, Kun Zhang, Anton van den Hengel, and Javen Qinfeng Shi. Analytic dag constraints for differentiable dag learning. arXiv preprint arXiv:2503.19218, 2025.  
+[30] Jiongzhi Zheng, Kun He, Jianrong Zhou, Yan Jin, and Chu-Min Li. Combining reinforcement learning with lin-kernighan-helsgaun algorithm for the traveling salesman problem. In Proceedings of the AAAI conference on artificial intelligence, volume 35, pages 12445-12452, 2021.  
+[31] Zhi Zheng, Changliang Zhou, Tong Xialiang, Mingxuan Yuan, and Zhenkun Wang. Udc: A unified neural divide-and-conquer framework for large-scale combinatorial optimization problems. arXiv preprint arXiv:2407.00312, 2024.
+
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+
+# Appendix: Solving Asymmetric Traveling Salesman Problem via Trace-Guided Cost Augmentation
+
+# A Linear Assignment Problem
+
+The linear assignment problem aims to determine a permutation matrix that minimizes the total assignment cost given by a cost matrix  $\mathbf{D} \in \mathbb{R}^{n \times n}$ , whose entries are denoted by  $d_{ij}$ . Let  $\mathbf{X} \in \mathbb{R}^{n \times n}$  be the assignment matrix, where  $x_{ij} = 1$  indicates that row  $i$  is assigned to column  $j$ , and zero otherwise. The problem can be formulated as the following linear program:
+
+$$
+\text {m i n i m i z e} \quad \sum_ {i = 1} ^ {n} \sum_ {j = 1} ^ {n} d _ {i j} x _ {i j}
+$$
+
+$$
+\text {s u b j e c t} \quad \sum_ {j = 1} ^ {n} x _ {i j} = 1 \quad \text {f o r a l l} i = 1, \dots , n, \tag {8}
+$$
+
+$$
+\sum_ {i = 1} ^ {n} x _ {i j} = 1 \quad \text {f o r a l l} j = 1, \dots , n,
+$$
+
+$$
+x _ {i j} \geq 0 \quad \text {f o r a l l} i, j.
+$$
+
+The dual problem introduces variables  $u_{i}$  and  $v_{j}$  corresponding to the row and column constraints, respectively, and is given by:
+
+$$
+\text {m a x i m i z e} \quad \sum_ {i = 1} ^ {n} u _ {i} + \sum_ {j = 1} ^ {n} v _ {j} \tag {9}
+$$
+
+$$
+\text {s u b j e c t} \quad u _ {i} + v _ {j} \leq d _ {i j} \quad \text {f o r a l l} i, j.
+$$
+
+Strong duality holds for this problem, guaranteeing that the optimal values of the primal and dual problems are equal when a feasible solution exists.
+
+The Hungarian Algorithm is, in fact, a primal-dual method for solving the Linear Assignment Problem (LAP). This insight allows us to exploit the structure of Algorithm 1, where multiple LAPs are solved iteratively. Specifically, we can retain the previously optimal dual variables  $\mathbf{u}$  and  $\mathbf{v}$  and use them as the initialization for subsequent runs of the Hungarian Algorithm. This warm-start strategy significantly accelerates convergence, as the optimal dual variables often remain close across successive LAP instances.
+
+# B Further Refining Performance by Branch-and-Bound
+
+We further refined our implementation by incorporating two key strategies: (1) reusing the dual variables  $\mathbf{u}$  and  $\mathbf{v}$ , as described in the previous section, and (2) introducing a simple branch-and-bound heuristic. In our branch-and-bound strategy, we first solve the Linear Assignment Problem (LAP) to obtain a lower bound for the ATSP instance. We then identify the edge with the highest cost in the LAP solution and branch on it. For each branch, we apply Algorithm 1 to compute a potentially global upper bound, while the LAP provides a local lower bound. Our heuristic allows up to five branch-and-bound iterations.
+
+With these enhancements, the results on large-scale ATsP instances satisfying the triangle inequality are presented in Table 8. As shown, reusing the dual variables accelerates our algorithm by up to a factor of five compared to the original implementation. Additionally, the branch-and-bound strategy offers further performance improvements. Similar gains are observed on purely random ATsP instances, as shown in Table 9.
+
+<table><tr><td rowspan="2">Method</td><td colspan="3">TSP500</td><td colspan="3">TSP1000</td></tr><tr><td>Avg. GAP</td><td>Median GAP</td><td>Runtime (s)</td><td>Avg. GAP</td><td>Median GAP</td><td>Runtime (s)</td></tr><tr><td>LKH-3 (10, 10000)</td><td>0.00%</td><td>0.00%</td><td>20.88</td><td>0.00%</td><td>0.00%</td><td>36.72</td></tr><tr><td>LKH-3 (1, 500)</td><td>1.08%</td><td>1.00%</td><td>1.02</td><td>51.23%</td><td>51.48%</td><td>3.93</td></tr><tr><td>LKH-3 (1, 10000)</td><td>0.25%</td><td>0.24%</td><td>2.93</td><td>3.25%</td><td>2.94%</td><td>7.05</td></tr><tr><td>MatPOENet</td><td>112.48%</td><td>112.19%</td><td>1.50</td><td>156.48%</td><td>156.24%</td><td>10.21</td></tr><tr><td>GOAL</td><td>1.07%</td><td>1.01%</td><td>6.85</td><td>1.26%</td><td>1.22%</td><td>58.48</td></tr><tr><td>Ours</td><td>-0.06 %</td><td>-0.11%</td><td>0.74</td><td>-0.35%</td><td>-0.36%</td><td>5.10</td></tr><tr><td>Ours-reuse[u, v]</td><td>-0.04%</td><td>-0.09%</td><td>0.19</td><td>-0.35%</td><td>-0.36%</td><td>0.97</td></tr><tr><td>Ours-reuse[u, v]-BnB</td><td>-0.11%</td><td>-0.12%</td><td>0.70</td><td>-0.38%</td><td>-0.38%</td><td>2.91</td></tr></table>
+
+Table 8: Comparison of methods on large-scale random ATSP with triangle inequality problems (TSP500 and TSP1000). GAP is measured relative to LKH-3 (10, 10000). The results are average from 128 instances. Our approaches outperforms the state-of-the-arts LKH-3 heuristics in terms of solution quality.  
+
+<table><tr><td rowspan="2">Method</td><td colspan="3">ATSP500</td><td colspan="3">ATSP1000</td></tr><tr><td>Avg. GAP</td><td>Median GAP</td><td>Runtime (s)</td><td>Avg. GAP</td><td>Median GAP</td><td>Runtime (s)</td></tr><tr><td>LKH-3 (10, 10000)</td><td>0.00%</td><td>0.00%</td><td>29.99</td><td>0.00%</td><td>0.00%</td><td>46.64</td></tr><tr><td>LKH-3 (1, 500)</td><td>24.47%</td><td>24.86%</td><td>1.08</td><td>51.23%</td><td>51.48%</td><td>3.81</td></tr><tr><td>LKH-3 (1, 10000)</td><td>2.50%</td><td>2.45%</td><td>3.85</td><td>3.25%</td><td>2.94%</td><td>7.91</td></tr><tr><td>MatPOENet</td><td>6400.51%</td><td>6400.50%</td><td>1.51</td><td>15017.16%</td><td>15016.70%</td><td>10.24</td></tr><tr><td>GOAL</td><td>369.69%</td><td>369.40%</td><td>6.84</td><td>305.84%</td><td>304.62%</td><td>58.53</td></tr><tr><td>Ours</td><td>-10.77 %</td><td>-11.59%</td><td>0.87</td><td>-21.40%</td><td>-21.86%</td><td>5.22</td></tr><tr><td>Ours-reuse[u,v]</td><td>-10.89%</td><td>-11.67%</td><td>0.33</td><td>-21.49%</td><td>-21.98%</td><td>1.54</td></tr><tr><td>Ours-reuse[u,v]-BnB</td><td>-11.88%</td><td>-12.02%</td><td>0.84</td><td>22.31%</td><td>-22.39%</td><td>3.63</td></tr></table>
+
+Table 9: Comparison of methods on large-scale pure random ATSP problems (TSP500 and TSP1000). GAP is measured relative to LKH-3 (10, 10000). The results are average from 128 instances. Our approaches outperforms the state-of-the-arts LKH-3 heuristics in terms of solution quality. The performance of learning based approaches is far worse than their reported results as they are trained on ATSP with triangle inequality and they failed to generalize to random ATSP.
+
+# C Experimental Results on TSPLIB instances
+
+We evaluated our algorithm on large-scale ATSP benchmark instances from TSPLIB, particularly those derived from stacker crane problems. Our method consistently outperforms LKH-3 (1-100) in terms of solution quality while also requiring less computational time. These results demonstrate the practical effectiveness of our approach on real-world structured benchmarks, further validating its scalability and competitiveness beyond synthetic instances.
+
+# D Comparison with ILP Solver
+
+We conducted a preliminary comparison between our method and an exact ILP solver on both symmetric and asymmetric TSP instances. Specifically, we tested the solver on 100-node symmetric problems and 500-node asymmetric problems and found that the ILP solver was able to return optimal solutions within a reasonable time budget.
+
+Due to time constraints, we evaluated both methods on 10 randomly sampled instances for each case. The results are summarized below. While the ILP solver guarantees optimality, our method achieves competitive solution quality with significantly faster runtimes, particularly for larger asymmetric instances—highlighting its scalability and practical utility for large-scale problems.
+
+# D.1 Performance comparison on 100-node symmetric TSP problems
+
+# D.2 Performance comparison on 500-node asymmetric TSP problems
+
+Table 10: Comparison of algorithms on asymmetric TSP benchmark instances.  
+
+<table><tr><td>Instance</td><td>Algorithm</td><td>Best Known Result</td><td>Running Time (s)</td><td>Result</td></tr><tr><td rowspan="5">rbg323</td><td>LKH(1-100)</td><td rowspan="5">1326</td><td>1.46</td><td>1388</td></tr><tr><td>LKH(1-10000)</td><td>3.38</td><td>1346</td></tr><tr><td>LKH(10-10000)</td><td>20.32</td><td>1346</td></tr><tr><td>Ours</td><td>0.42</td><td>1360</td></tr><tr><td>Ours (no 2-opt)</td><td>0.20</td><td>1365</td></tr><tr><td rowspan="5">rbg358</td><td>LKH(1-100)</td><td rowspan="5">1163</td><td>1.67</td><td>1294</td></tr><tr><td>LKH(1-10000)</td><td>4.03</td><td>1175</td></tr><tr><td>LKH(10-10000)</td><td>25.16</td><td>1175</td></tr><tr><td>Ours</td><td>0.09</td><td>1180</td></tr><tr><td>Ours (no 2-opt)</td><td>0.04</td><td>1180</td></tr><tr><td rowspan="5">rbg403</td><td>LKH(1-100)</td><td rowspan="5">2465</td><td>6.27</td><td>2536</td></tr><tr><td>LKH(1-10000)</td><td>26.99</td><td>2498</td></tr><tr><td>LKH(10-10000)</td><td>9.06</td><td>2498</td></tr><tr><td>Ours</td><td>1.53</td><td>2473</td></tr><tr><td>Ours (no 2-opt)</td><td>1.28</td><td>2473</td></tr><tr><td rowspan="5">rbg443</td><td>LKH(1-100)</td><td rowspan="5">2720</td><td>5.44</td><td>2813</td></tr><tr><td>LKH(1-10000)</td><td>7.87</td><td>2762</td></tr><tr><td>LKH(10-10000)</td><td>30.06</td><td>2756</td></tr><tr><td>Ours</td><td>0.52</td><td>2760</td></tr><tr><td>Ours (no 2-opt)</td><td>0.27</td><td>2760</td></tr></table>
+
+Table 11: Performance comparison on 100-node symmetric TSP problems.  
+
+<table><tr><td>Algorithm</td><td>Running Time (s)</td><td>Ratio to Optimal (Mean)</td><td>Ratio to Optimal (Median)</td></tr><tr><td>LKH(1–100)</td><td>0.07</td><td>0.61%</td><td>0.17%</td></tr><tr><td>LKH(1–10000)</td><td>1.24</td><td>0.12%</td><td>0.00%</td></tr><tr><td>LKH(10–10000)</td><td>12.14</td><td>0.00%</td><td>0.00%</td></tr><tr><td>Gurobi</td><td>2.06</td><td>0.00%</td><td>0.00%</td></tr><tr><td>Ours (no local search)</td><td>0.03</td><td>15.2%</td><td>14.4%</td></tr><tr><td>Ours (local search)</td><td>0.05</td><td>1.47%</td><td>1.35%</td></tr></table>
+
+Table 12: Performance comparison on 500-node asymmetric TSP problems.  
+
+<table><tr><td>Algorithm</td><td>Running Time (s)</td><td>Ratio to Optimal (Mean)</td><td>Ratio to Optimal (Median)</td></tr><tr><td>LKH(1-100)</td><td>0.87</td><td>2.22%</td><td>0.17%</td></tr><tr><td>LKH(1-10000)</td><td>2.82</td><td>0.45%</td><td>0.00%</td></tr><tr><td>LKH(10-10000)</td><td>20.23</td><td>0.18%</td><td>0.00%</td></tr><tr><td>Gurobi</td><td>6.36</td><td>0.00%</td><td>0.00%</td></tr><tr><td>Ours (no local search)</td><td>0.17</td><td>0.005%</td><td>0.001%</td></tr><tr><td>Ours (local search)</td><td>0.30</td><td>0.005%</td><td>0.001%</td></tr></table>

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NeurIPS/2025/SongBloom_ Coherent Song Generation via Interleaved Autoregressive Sketching and Diffusion Refinement/full.md

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+# SongBloom: Coherent Song Generation via Interleaved Autoregressive Sketching and Diffusion Refinement
+
+Chenyu Yang $^{1,4}$
+
+Shuai Wang $^{3,4,*}$
+
+Hangting Chen
+
+Wei Tan²
+
+Jianwei Yu
+
+Haizhou Li $^{1,4,*}$
+
+<sup>1</sup>The Chinese University of Hong Kong, Shenzhen <sup>2</sup>Tencent AI Lab
+
+$^{3}$ Nanjing University  $^{4}$ Shenzhen Research Institution of Big Data
+
+# Abstract
+
+Generating music with coherent structure, harmonious instrumental and vocal elements remains a significant challenge in song generation. Existing language models and diffusion-based methods often struggle to balance global coherence with local fidelity, resulting in outputs that lack musicality or suffer from incoherent progression and mismatched lyrics. This paper introduces SongBloom<sup>1</sup>, a novel framework for full-length song generation that leverages an interleaved paradigm of autoregressive sketching and diffusion-based refinement. SongBloom employs an autoregressive diffusion model that combines the high fidelity of diffusion models with the scalability of language models. Specifically, it gradually extends a musical sketch from short to long and refines the details from coarse to fine-grained. The interleaved generation paradigm effectively integrates prior semantic and acoustic context to guide the generation process. Experimental results demonstrate that SongBloom outperforms existing methods across both subjective and objective metrics and achieves performance comparable to the state-of-the-art commercial music generation platforms. Audio samples are available on our demo page: https://cypress-yang.github.io/SongBloom_demo.
+
+# 1 Introduction
+
+Music, as one of the most expressive forms of human art, serves as a universal language that transcends cultural and linguistic boundaries. Lyric-to-song generation, which involves producing both vocals and accompaniment from given lyrics, requires comprehensive modeling of diverse musical elements, including instrumentation, vocal expressiveness, structural arrangement, and emotional dynamics. These complexities pose significant challenges for existing generative approaches [1, 2]. In particular, end-to-end models that directly synthesize music face considerable challenges in maintaining sound quality, as music's wide frequency range and complex temporal dynamics impose stringent demands on generative fidelity and model capacity.
+
+Advanced approaches to long-form generation typically adopt either a unified non-autoregressive (NAR) architecture [3] that models the entire process with a diffusion transformer, or an autoregressive (AR) framework [4, 5] that employs language models (LMs) and separates the generation into semantic and acoustic stages. However, NAR models often struggle to capture precise alignments between phonemes and audio frames, whereas AR approaches typically predict quantized tokens that suffer from low bitrates, resulting in weak musicality and degraded audio quality.
+
+<table><tr><td>System</td><td>Architecture</td><td>Size</td><td>Max Length</td><td>Sample Rate (kHz)</td><td>Structure</td></tr><tr><td>SongGen [9]</td><td>LM</td><td>1.1B</td><td>30s</td><td>16</td><td>×</td></tr><tr><td>SongEditor [5]</td><td>LM + Diff.</td><td>0.7B + 1B</td><td>120s</td><td>44.1</td><td>✓</td></tr><tr><td>DiffRhythm-base [3]</td><td>Diff.</td><td>1.1B</td><td>95s</td><td>44.1</td><td>×</td></tr><tr><td>DiffRhythm-full [3]</td><td>Diff.</td><td>1.1B</td><td>285s</td><td>44.1</td><td>×</td></tr><tr><td>YuE [4]</td><td>LM + LM</td><td>7B + 1B</td><td>300s</td><td>44.1</td><td>✓</td></tr><tr><td>SongBloom-tiny</td><td>LM &amp; Diff.</td><td>1.3B</td><td>60s</td><td>48</td><td>✓</td></tr><tr><td>SongBloom-full</td><td>LM &amp; Diff.</td><td>2B</td><td>150s</td><td>48</td><td>✓</td></tr></table>
+
+Table 1: Comparison of existing song generation models. “+” indicates two cascaded models, while “&” indicates one unified model with jointly optimized objectives.
+
+To address these challenges, we propose SongBloom, a novel lyric-to-song generation approach that combines the strengths of both AR and NAR approaches. SongBloom is built upon an autoregressive diffusion architecture [6], which is well-suited for continuous-valued modalities such as music and speech, and has shown strong scalability and robust long-range generation capabilities [7] comparable to that of audio language models. Given structured lyrics and a 10-second reference audio clip, SongBloom can generate full-length songs with diverse sections up to 150 seconds long.
+
+To further improve the semantic alignment in long-form song generation, we decouple high-level sketch planning from low-level acoustic synthesis, and integrate both within the unified autoregressive framework that supports joint modeling in the style of Chain-of-Thought (CoT) prompting [8]. Notably, we observe that the entire sketch sequence is not always necessary for predicting each acoustic frame. Instead, the acoustic context itself can provide valuable guidance for shaping subsequent sketch planning. Based on this insight, we partition both semantic and acoustic sequences into patches and generate them in an interleaved manner. This paradigm not only facilitates bidirectional contextual exchange between sketching and refinement stages but also significantly reduces the sequence length needed during acoustic synthesis.
+
+Both subjective and objective evaluations are conducted in our experiments. The assessment employs a comprehensive suite of metrics covering multiple dimensions, including musicality, audio quality, structural coherence, and overall musical aesthetics. The results show that SongBloom consistently outperforms all open-source baselines and surpasses most commercial systems, achieving state-of-the-art performance across several key metrics.
+
+The main contributions of this paper can be concluded as:
+
+- We propose SongBloom, the first autoregressive diffusion-based model for full-length song generation, which is capable of producing high-quality and expressive songs with coherent structure and rich acoustic details.  
+- We design a unified framework that combines coarse and fine stages into a single, jointly optimized model, and treats sketch tokens as CoT-like prompts for acoustic generation directly. Additionally, we introduce a novel interleaved generation paradigm that alternates between sketch and acoustic patches, effectively utilizing both semantic and acoustic contexts.  
+- Experiments demonstrate that SongBloom surpasses all non-commercial baselines and achieves competitive performance with the state-of-the-art Suno-v4.5 in terms of both subjective and objective metrics. Additionally, our approach achieves a relatively low real-time factor (RTF) compared to other LM-based methods, indicating a favorable trade-off between generation quality and computational efficiency.
+
+# 2 Related work
+
+# 2.1 Song Generation
+
+Song generation models aim to produce both singing voice and accompanying music. Previous studies, such as Melodist [10], SongGen [9], and MelodyLM [11], have shown promising results in generating sentence-level singing and accompaniment. However, generating long-form song pieces that encompass diverse musical structures, such as verses, choruses, and instrumental-only sections,
+
+remains a significant challenge. JukeBox [12] was one of the earliest attempts at generating long-form songs, though it suffers from limited genre and timbre control. SongComposer [13] supports long-form lyric-to-melody generation but struggles to preserve certain aspects such as timbre.
+
+Recent research has increasingly focused on long-form and structure-guided song generation. DiffRhythm [3] introduces a single diffusion model capable of generating songs up to 285 seconds in length while maintaining a low real-time factor. YuE [4] employs a coarse-to-fine generation strategy, leveraging two large-scale language models to produce acoustic tokens with improved fidelity and structure. SongEditor [5] proposes a two-stage framework that supports both full song generation and infilling-based editing tailored to specific lyrics, offering greater flexibility for user-driven customization. In addition, commercial platforms such as  $\mathrm{Sun}^{2}$  and  $\mathrm{Ud}^{3}$  have demonstrated promising capabilities in complete song generation. However, due to the lack of publicly available technical details, these systems cannot be thoroughly evaluated or compared.
+
+# 2.2 Autoregressive Diffusion Models
+
+While language models predominantly operate in discrete-valued space, this paradigm might be suboptimal for inherently continuous audio signals. A common workaround involves transforming waveforms into discrete token sequences through vector quantization (VQ)[14, 15] or residual vector quantization (RVQ)[16]. Recent work, however, challenges the necessity of discretization. For instance, Li et al. [6] demonstrates that discrete tokens may not be essential for autoregressive language models in image generation. Some research has also explored autoregressive diffusion techniques for text [17] and speech [18] generation. ARDiT [19] introduces a decoder-only diffusion transformer for zero-shot text-to-speech generation, achieving impressive results without relying on discrete tokens. DiTAR [7] further simplifies the architecture through a divide-and-conquer strategy, making the model more adaptable for large-scale deployment.
+
+However, the approaches mentioned above remain restricted to the speech domain, and direct migration faces some limitations. First, songs are typically longer than spoken utterances, imposing stricter requirements on contextual consistency across extended durations. Second, the presence of musical accompaniment leads to a higher semantic density of each frame, making generation more difficult. Recent works [20, 21] introduce a shared transformer architecture capable of handling both discrete and continuous tokens across different modalities. Inspired by this design, we propose a unified autoregressive framework that integrates both sketch prediction and audio synthesis with distinct training objectives.
+
+# 3 Preliminary
+
+Next-Token Prediction Given a sequence of tokens  $\mathbf{x} = (x_{1}, x_{2}, \ldots, x_{t - 1})$ , the objective is to model the conditional probability of the next token  $P(x_{t} \mid x_{1}, x_{2}, \ldots, x_{t - 1})$ . For an entire sequence  $(x_{1}, x_{2}, \ldots, x_{T})$ , the joint probability is factorized as:
+
+$$
+P \left(x _ {1}, x _ {2}, \dots , x _ {T} \mid C\right) = \prod_ {t = 1} ^ {T} P \left(x _ {t} \mid x _ {1}, x _ {2}, \dots , x _ {t - 1}; C\right), \tag {1}
+$$
+
+where  $C$  represents the conditions.
+
+Rectified Flow Matching Rectified flow-matching [22] defines a linear interpolation between a source point  $z_0$  and a target point  $z_1$  over time  $t \in [0,1]$ :
+
+$$
+z _ {t} = (1 - t) z _ {0} + t z _ {1} \tag {2}
+$$
+
+$$
+\mathrm {d} z _ {t} = \left(z _ {1} - z _ {0}\right) \mathrm {d} t = v \left(z _ {t}, t\right) \mathrm {d} t \tag {3}
+$$
+
+where  $z_0$  denotes the original data and  $z_1 \sim \mathcal{N}(0,1)$  is a random noise.  $v(z_t,t)$  represents the velocity field guiding the transformation from  $z_0$  to  $z_1$ . The reverse process enables the generation by inverting this transformation:
+
+$$
+z _ {t - \Delta t} = z _ {t} - v \left(z _ {t}, t\right) \cdot \Delta t \tag {4}
+$$
+
+This formulation allows the model to sample from the data distribution by iteratively denoising from the noise distribution.
+
+![](images/49bb68850263bb26feca8cce5095d61e45404723588cb4d6b2e1abe07487c2ec.jpg)  
+Figure 1: Overall architecture of SongBloom.
+
+# 4 SongBloom
+
+# 4.1 Task Formulation
+
+MusicLM [23] proposes a two-stage framework that leverages semantic tokens, referred to as sketch, extracted from self-supervised learning (SSL) models such as MERT [24] as intermediate representations. In the first stage, an LM is employed to generate the sketch sequence from scratch, as its variable length and discrete nature make it well-suited for language modeling approaches, which have demonstrated strong scalability. In the second stage, either diffusion or LM subsequently generates acoustic tokens in a coarse-to-fine manner. The overall generation process can be formulated as:
+
+$$
+p \left(a _ {(0: T ]}, s _ {(0: T ]} \mid C\right) = p _ {\theta} \left(a _ {(0: T ]} \mid s _ {(0: T ]}, C\right) \cdot p _ {\phi} \left(s _ {(0: T ]} \mid C\right), \tag {5}
+$$
+
+where  $s_i$  denotes the sketch token at frame  $i$ , and  $a_i$  denotes the acoustic token.  $T$  is the total length of the frame sequence. The parameters  $\phi$  and  $\theta$  represent the models responsible for generating semantic and acoustic tokens, respectively. This two-stage architecture has been widely adopted in previous long-form music generation approaches such as [5, 4, 25].
+
+However, the aforementioned formulation faces several limitations: 1) Future sketch tokens contribute little to the prediction of current acoustic tokens due to the strict token-level alignment between the two representations. As noted in Yang et al. [5], Xu et al. [26], using chunk-level semantic-to-acoustic reconstruction introduces minimal degradation in fluency, suggesting that generating the complete semantic sequence in advance may be unnecessary. 2) Given the rich and expressive nature of music, prior acoustic latents can provide valuable contextual cues for sketch token generation. However, existing approaches typically condition only on the semantic history, thereby overlooking potentially useful acoustic information.
+
+To address the aforementioned challenges, we propose a novel generation paradigm that interleaves the generation of sketch and acoustic sequences. Both types of tokens are first segmented into fixed-size patches. The modified generation process is formulated as:
+
+$$
+p \left(a _ {(0: T ]}, s _ {(0: T ]} \mid C\right) = \prod_ {i = 0} ^ {N} p _ {\theta} \left(s _ {(i P: (i + 1) P ]} \mid s _ {(0: i P ]}, a _ {(0: i P ]}, C\right) \cdot p _ {\phi} \left(a _ {(i P: (i + 1) P ]} \mid s _ {(0: (i + 1) P ]}, a _ {(0: i P ]}, C\right), \tag {6}
+$$
+
+where  $P$  denotes the patch size,  $N = \lceil T / P\rceil -1$  denotes the number of patches.  $\theta$  and  $\phi$  correspond to the parameters responsible for generating sketch tokens and acoustic features, respectively. Although a sequential dependency between the two stages still exists, the interleaved generation paradigm enables bidirectional information exchange between the semantic and acoustic representations. The two stages share a subset of model parameters and are jointly optimized within a unified framework, facilitating coherent and high-fidelity song generation.
+
+# 4.2 Data Representation
+
+Lyric preprocessing To incorporate structural information, we introduce two categories of structure flags into the lyric sequence. Vocal-based flags, which indicate the structure of vocal sections (e.g., verse, chorus), are prepended to the beginning of each paragraph. Accompaniment-based flags, which represent non-vocal regions such as intros and outros, are inserted between paragraphs and expanded proportionally according to their exact durations. All lyric text is normalized and then transformed into a phoneme sequence to serve as input for the subsequent sketch generation stage.
+
+Sketch tokens In this paper, we adopt the embeddings extracted from MuQ [27] as our sketch representation. A single vector quantization layer is attached to discretize the sketch embeddings into flattened tokens. Additionally, we have explored various sketch alternatives including musical signals, which are further discussed in Appendix A.
+
+Acoustic latents Due to the broader frequency spectrum of music compared to speech, modeling and reconstruction become significantly more challenging. Codec-based discretization methods [28, 29] inherently suffer from a trade-off between reconstruction fidelity and the depth of codebooks, often increasing generation complexity. To tackle this issue, we substitute the discrete acoustic tokens with continuous latents derived from an autoencoder, which compresses 2-channel  $48\mathrm{kHz}$  music into continuous-valued acoustic latent sequences at a reduced frame rate. This continuous representation preserves high-frequency detail while simplifying the generation process, making it more suitable for high-fidelity music synthesis.
+
+# 4.3 Architecture
+
+# 4.3.1 Overview
+
+The overall model architecture is illustrated in Figure 1, which consists of three main modules: an autoregressive transformer decoder, a non-autoregressive diffusion transformer, and an acoustic encoder. Both semantic and acoustic sequences are first segmented into fixed-length patches and then generated in an interleaved manner, enabling the model to maintain consistency between high-level structure and fine-grained acoustic detail.
+
+# 4.3.2 Autoregressive Sketch Generation
+
+The autoregressive sketch generation stage is designed to generate both sketch tokens  $p_{\theta}(s_{(iP:(i + 1)P]}\mid \cdot)$  and a corresponding hidden vector  $p_{\theta}(h_i|\cdot)$  per patch, which serve as conditioning inputs for the downstream diffusion stage. The architecture is composed of a stack of transformer decoders with causal masks, enabling left-to-right autoregressive prediction over the sketch tokens. Conditions, including lyric text and style prompt, are prepended to the semantic stream. Each semantic patch is generated based on all previously generated sketch tokens, as well as the acoustic latent features from preceding patches. Specifically, the acoustic features of the current patch are compressed via an acoustic encoder and then inserted as the next token at the position of the hidden vector. This design eliminates the need for custom attention masks, thereby facilitating compatibility with acceleration techniques such as FlashAttention2 [30].
+
+The training objective for the autoregressive sketch generation is formulated as a cross-entropy loss:
+
+$$
+\mathcal {L} _ {\mathrm {L M}} = - \frac {1}{N P} \sum_ {i = 0} ^ {N - 1} \sum_ {j = 1} ^ {P} \log p _ {\theta} \left(s _ {i P + j} \mid s _ {<   i P + j}, a _ {<   i P}, C\right), \tag {7}
+$$
+
+where  $s_t$  denotes the sketch token at timestep  $t$ .
+
+# 4.3.3 Non-Autoregressive Latent Diffusion
+
+The non-autoregressive diffusion module predicts acoustic latents within each patch in parallel using a full-attention DiT [31] architecture. The model is trained with the Rectified Flow-Matching (RFM) objective [22], which aims to predict the velocity field  $v_{\phi}(\cdot)$  governing the dynamics of the latent trajectories:
+
+$$
+\frac {\mathrm {d} \mathbf {z} _ {t} ^ {i}}{\mathrm {d} t} = v _ {\phi} \left(t, \mathbf {z} _ {t} ^ {i} \mid h _ {i}, s _ {(i P: (i + 1) P ]}, \mathbf {z} _ {0} ^ {i - 1}\right). \tag {8}
+$$
+
+Here,  $t$  denotes the diffusion time step. In our experiments, we sample  $t$  from a logit-normal distribution  $\pi_t$  following Esser et al. [32].  $\mathbf{z}_t^i$  represents the latent variables at time  $t$  for the  $i$ -th patch, where  $\mathbf{z}_0^i = a_{(iP:(i + 1)P]}$  and  $\mathbf{z}_1^i\sim \mathcal{N}(\mathbf{0},\mathbf{I})$ . The variable  $h_i$  is the hidden vector inherited from the first stage.  $s_{(iP:(i + 1)P]}$  refers to the sketch tokens for the current patch, and  $\mathbf{z}_0^{i - 1}$  refers to the acoustic latents from the preceding patch.
+
+The loss function for training is defined as:
+
+$$
+\mathcal {L} _ {\text {f l o w}} = \mathbb {E} _ {t \sim \pi_ {t}, \mathbf {z} ^ {i}} \left[ | | v _ {\phi} (t, \mathbf {z} _ {t} ^ {i} \mid \cdot) - (\mathbf {z} _ {1} ^ {i} - \mathbf {z} _ {0} ^ {i}) | | ^ {2} \right], \tag {9}
+$$
+
+where  $\pi_t$  denotes the distribution over diffusion steps. The total training objective combines the sketch generation loss and the diffusion loss:
+
+$$
+\mathcal {L} = \mathcal {L} _ {\mathrm {L M}} + \lambda \cdot \mathcal {L} _ {\text {f l o w}}, \tag {10}
+$$
+
+where  $\lambda = 0.1$  is an empirically chosen weighting factor.
+
+Gradients are backpropagated from the diffusion stage to the sketch generation stage via the hidden vector  $h_i$ . Additionally, the sketch tokens share the same embedding layer across both stages, which improves training efficiency and accelerates convergence.
+
+Jia et al. [7] introduced a classifier-free guidance (CFG) mechanism for autoregressive diffusion models, which is applied only to the hidden vector. In our work, we treat the hidden vector and sketch tokens as a unified conditioning signal, and jointly mask them during training. Additional masks are applied solely to the sketch tokens to further encourage the hidden vector to carry more semantic information.
+
+# 5 Experimental Setup
+
+# 5.1 Dataset
+
+We use a large-scale song dataset totaling 100K hours, including both Chinese and English songs, to train our model. The original lyrics are of relatively low quality, prompting the use of a dedicated data cleaning pipeline. First, we separate the vocal and instrumental tracks using the Demucs model [33]. Then we use the WhisperX toolkit [34] to filter out mismatched lyric sentences and recover missing words. The time boundaries of lyrics are also refined based on the results of automatic speech recognition (ASR). Additionally, structural information is extracted from the original tracks using the structure analyzer proposed by Kim and Nam [35].
+
+For inference, considering that most current song generation models are trained on large-scale but closed-source datasets, it is crucial to ensure that the test samples are not seen during training. To this end, the lyrics are generated using GPT-4o, and the reference audio is randomly clipped from unseen real songs with diverse genres, followed by random shuffling to prevent potential data leakage. A subset comprising 20 samples is used for subsequent human evaluation.
+
+# 5.2 Configuration & Training Setup
+
+In our experiments, the codebook size for semantic tokens is set to 16384, at a frame rate of 25. The stable-audio-vae [36] is adopted as the implementation of our waveform autoencoder. Minor modifications are made to its hyperparameters to ensure they have the same frame rate, thereby facilitating synchronized sketches and latents.
+
+The core component of SongBloom is based on the LLaMA-2 decoder architecture [37], utilizing causal attention as the backbone of the autoregressive LM. This architecture is further modified to support bidirectional attention, forming our diffusion transformer. Rotary Positional Embeddings (RoPE) [38] are employed in both autoregressive and non-autoregressive transformers to encode positional information. The acoustic encoder is a simple two-layer convolutional network. All conditioning inputs are prepended to the input sequence. Attention modules in each layer use 24 heads with a hidden dimension of 1536, consistent across both autoregressive and non-autoregressive components. The patch size is set to 16, spanning 0.64 seconds. We evaluate two model configurations in our experiments: (1) SongBloom-tiny, comprising 16 layers for the autoregressive LM and 8 layers for the non-autoregressive diffusion transformer, capable of generating songs up to 60 seconds in length; and (2) SongBloom-full, comprising 24 autoregressive layers and 12 non-autoregressive
+
+Table 2: Objective evaluation results across all models. The upper section reports the performance of commercial platforms (valid until June 23, 2025), while the lower section presents results for open-source baselines as well as our proposed models.  
+
+<table><tr><td rowspan="2">Models</td><td rowspan="2">Prompt</td><td rowspan="2">PER(%)↓</td><td rowspan="2">MCC↑</td><td rowspan="2">FAD↓</td><td rowspan="2">SER(%)↓</td><td colspan="4">Aesthetic Score</td><td rowspan="2">RTF↓</td></tr><tr><td>CE↑</td><td>CU↑</td><td>PC↑</td><td>PQ↑</td></tr><tr><td>Suno-v4.5 †</td><td>text</td><td>24.67</td><td>0.69</td><td>3.39</td><td>10.43</td><td>7.77</td><td>7.93</td><td>6.03</td><td>8.40</td><td>-</td></tr><tr><td>Udio-v1.5</td><td>10s wav</td><td>20.04</td><td>0.79</td><td>4.04</td><td>17.92</td><td>7.47</td><td>7.63</td><td>6.29</td><td>8.20</td><td>-</td></tr><tr><td>Haimian</td><td>text</td><td>10.03</td><td>0.63</td><td>5.45</td><td>13.39</td><td>7.55</td><td>7.87</td><td>5.75</td><td>8.28</td><td>-</td></tr><tr><td>Mureka-O1</td><td>10s wav</td><td>7.79</td><td>0.86</td><td>3.39</td><td>31.37</td><td>7.69</td><td>7.84</td><td>6.41</td><td>8.45</td><td>-</td></tr><tr><td>ACE-step [46]</td><td>text</td><td>54.34</td><td>0.62</td><td>8.02</td><td>12.27</td><td>7.37</td><td>7.52</td><td>6.26</td><td>7.85</td><td>-</td></tr><tr><td>YuE-7B [4]</td><td>text + 30s wav</td><td>27.30</td><td>0.62</td><td>5.99</td><td>13.93</td><td>7.25</td><td>7.59</td><td>5.96</td><td>8.03</td><td>13.724</td></tr><tr><td>DiffRhythm-full [3]‡</td><td>10s wav</td><td>15.77</td><td>0.70</td><td>5.04</td><td>46.62</td><td>5.81</td><td>7.29</td><td>4.52</td><td>7.73</td><td>0.034</td></tr><tr><td>SongEditor [5]</td><td>10s wav</td><td>16.20</td><td>0.77</td><td>4.85</td><td>18.06</td><td>7.44</td><td>7.80</td><td>6.06</td><td>8.27</td><td>1.717</td></tr><tr><td>SongBloom-full</td><td>10s wav</td><td>6.75</td><td>0.88</td><td>3.43</td><td>17.67</td><td>7.71</td><td>7.88</td><td>5.86</td><td>8.43</td><td>1.649</td></tr><tr><td>SongBloom-full-ft</td><td>10s wav</td><td>5.49</td><td>0.86</td><td>3.20</td><td>14.50</td><td>7.79</td><td>7.96</td><td>5.88</td><td>8.47</td><td>1.649</td></tr></table>
+
+$\dagger$ DiffRhythm requires additional sentence-level timestamps. We first extract paragraph-level timestamps from ASR results of generated samples, then employ GPT-4o to predict the start time of each sentence.
+
+layers, enabling song generation up to 150 seconds. The former is used for analysis and ablation studies, while the latter is included to enable fair comparisons with other baselines. The model is trained with 16 A100 GPUs for approximately one week.
+
+All models are trained using the AdamW optimizer [39] with a learning rate of 1e-4. A cosine learning rate scheduler [40] with 2000 warm-up steps is employed to stabilize early training. Each model is trained for approximately 150K steps with a batch size of 128. The DeepSpeed strategy [41] is adopted to support efficient training. For inference, both stages share a classifier-free guidance coefficient of 1.5. Next-token prediction is performed using top-k sampling with  $k = 200$  and a temperature of 0.9. The diffusion process employs the Euler ODE solver with 36 diffusion steps.
+
+# 5.3 Evaluation Metrics
+
+We evaluate the proposed models using the following objective metrics: (1) Phoneme Error Rate (PER), computed based on the separated vocal tracks and the corresponding lyrics; (2) MuLan Cycle Consistency (MCC), which measures the cosine similarity of MuLan [42] embeddings between generated samples and reference audio or textual descriptions; (3) Fréchet Audio Distance (FAD)[43], which quantifies the distributional similarity between generated samples and real songs from which the clips are intercepted; and (4) Structural Error Rate (SER), which measures the mismatch between detected structural patterns and the target lyric structure. The Dynamic Time Warping (DTW) algorithm [44] is first employed to obtain the optimal temporal alignment, and then we calculate the proportion of error duration. Additionally, we leverage the Audiobox-Aesthetic [45] to assess musical aesthetics, including content enjoyment (CE), content usefulness (CU), production complexity (PC), and production quality (PQ).
+
+For subjective evaluation, we conduct a Mean Opinion Score (MOS) listening test. A group of at least 10 participants with musical expertise are invited to rate each sample on a scale from 1 to 5. The evaluation focuses on several aspects: musicality (MUS) and audio quality (QLT) assessed separately for both the vocal and accompaniment components, correctness (CRR) of lyrics, and consistency (CST) between samples and prompts.
+
+# 6 Results
+
+# 6.1 Full-Length Lyric-to-Song Generation
+
+Based on the SongBloom-full model, we fine-tuned it for 1,000 additional steps on synthesized data with a clear alternating structure of verses and choruses, denoted as SongBloom-full-ft. This fine-tuning process enhances the model's ability to emulate synthesized lyric compositions, leading to improved performance on lyrics with similar stylistic patterns.
+
+Comparison of objective metrics As shown in Table 2, SongBloom demonstrates superior performance across various metrics and is competitive with Suno-v4.5, the state-of-the-art commercial song
+
+Table 3: Subjective evaluation results. Consistency (CST) is reported only for models that accept a 10-second reference audio as the style prompt.  
+
+<table><tr><td>Models</td><td>MUSV↑</td><td>MUSA↑</td><td>QLTV↑</td><td>QLTA↑</td><td>CRR↑</td><td>CST↑</td></tr><tr><td>Suno-v4.5</td><td>3.87 ± 0.28</td><td>4.04 ± 0.18</td><td>3.83 ± 0.12</td><td>3.96 ± 0.06</td><td>2.95 ± 0.14</td><td>-</td></tr><tr><td>Udio-v1.5</td><td>3.28 ± 0.27</td><td>3.55 ± 0.25</td><td>3.62 ± 0.23</td><td>3.74 ± 0.21</td><td>2.57 ± 0.19</td><td>2.76 ± 0.31</td></tr><tr><td>Haimian</td><td>3.39 ± 0.26</td><td>3.65 ± 0.24</td><td>3.86 ± 0.17</td><td>3.90 ± 0.11</td><td>3.09 ± 0.20</td><td>-</td></tr><tr><td>Mureka-O1</td><td>3.91 ± 0.26</td><td>3.93 ± 0.14</td><td>3.85 ± 0.18</td><td>3.89 ± 0.12</td><td>3.38 ± 0.16</td><td>3.41 ± 0.39</td></tr><tr><td>ACE-step [46]</td><td>2.95 ± 0.29</td><td>2.93 ± 0.32</td><td>2.66 ± 0.27</td><td>2.68 ± 0.25</td><td>2.11 ± 0.16</td><td>-</td></tr><tr><td>YuE-7B [4]</td><td>2.93 ± 0.27</td><td>3.15 ± 0.24</td><td>3.16 ± 0.27</td><td>3.28 ± 0.23</td><td>2.54 ± 0.19</td><td>-</td></tr><tr><td>DiffRhythm-full [3]</td><td>2.99 ± 0.27</td><td>3.48 ± 0.27</td><td>2.97 ± 0.21</td><td>3.33 ± 0.26</td><td>2.51 ± 0.23</td><td>2.45 ± 0.28</td></tr><tr><td>SongEditor [5]</td><td>2.90 ± 0.32</td><td>3.11 ± 0.27</td><td>3.03 ± 0.32</td><td>3.24 ± 0.28</td><td>2.89 ± 0.16</td><td>2.44 ± 0.28</td></tr><tr><td>SongBloom-full</td><td>3.59 ± 0.25</td><td>3.60 ± 0.23</td><td>3.83 ± 0.09</td><td>3.81 ± 0.08</td><td>3.27 ± 0.17</td><td>3.62 ± 0.27</td></tr><tr><td>SongBloom-full-ft</td><td>3.91 ± 0.24</td><td>3.92 ± 0.12</td><td>3.95 ± 0.04</td><td>3.93 ± 0.10</td><td>3.42 ± 0.18</td><td>3.45 ± 0.31</td></tr></table>
+
+generation platform. After fine-tuning on downstream data, SongBloom-full-ft even outperforms Suno-v4.5 in several metrics.
+
+During our evaluation, we observed that Suno tends to follow rigid structural patterns, such as redundantly repeating the chorus at the end, which leads to structural hallucinations and degraded PER performance. In contrast, SongBloom adheres more faithfully to the structure of the input lyrics, enabling flexible and structurally coherent song generation that significantly reduces the PER. In terms of the MCC metric, waveform-based style prompts generally possess a natural advantage over text descriptions, as they always provide a comprehensive and unbiased representation of all musical components. Among these approaches, SongBloom-full achieves the highest MCC score. In terms of the automated-evaluated aesthetic scores, SongBloom-full-ft outperforms all other baselines in three out of the four metrics, further demonstrating its ability to generate high-quality, musically coherent, and aesthetically pleasing songs.
+
+Apart from generation performance, we also assessed the inference speed of SongBloom compared to other open-source models. The integrated design of SongBloom enables outstanding inference efficiency, yielding a lower RTF than all other autoregressive baselines. Compared to SongEditor, our model achieves a similar RTF despite having a larger size, since SongEditor takes the entire semantic sequence as input during the diffusion stage, leading to unnecessary computational overhead. Meanwhile, YuE utilizes two LMs and generates multi-layer codec tokens in a flattened pattern, significantly extending the generation sequence. While autoregressive models are naturally slower than their non-autoregressive counterparts, SongBloom achieves an excellent trade-off between efficiency and performance.
+
+Comparison of subjective metrics Table 3 shows that SongBloom and Suno-v4.5 dominate the human evaluation results, outperforming all other models by a clear margin. Suno-v4.5 demonstrates particular strength in accompaniment generation, whereas SongBloom excels in metrics related to vocal tracks. Notably, SongBloom achieves the highest correctness score, indicating a stronger adherence to the provided lyrics compared to other systems. This suggests that its interleaved generation paradigm effectively preserves semantic intent throughout the song. Furthermore, SongBloom's superior performance in consistency is corroborated by its high MCC score, reflecting its ability to maintain thematic coherence over every section of songs.
+
+# 6.2 Ablation Study on Diffusion Conditions
+
+Table 4 compares the performance of SongBloom-tiny under different combinations of diffusion conditions. In the absence of sketch tokens, the LM stage produces only a sequence of hidden vectors, and the diffusion stage reconstructs the audio solely based on them. As shown in the table, the sketch plays a critical role in generation. Without it as a coarse-grained CoT, the model fails to learn proper alignment between phoneme sequences and audio, resulting in extremely high PER. In contrast, when the sketch is provided, even without additional conditions, the diffusion transformer can generate intelligible vocals solely based on the hidden vector from the LM stage. Meanwhile, incorporating both the acoustic context and sketch tokens as input conditions during the diffusion stage further enhances both objective and subjective performance, leading to higher-quality generation results.
+
+Table 4: Ablation study of SongBloom-tiny under different diffusion conditions. "H" represents the hidden vector, "C" represents the acoustic context from the previous patch, and "S" represents the sketch tokens of the current patch.  
+
+<table><tr><td rowspan="2">w/ sketch</td><td rowspan="2">Conditions</td><td colspan="2">Objective</td><td colspan="4">Aesthetic Score</td><td colspan="2">Subjective</td></tr><tr><td>PER(%)↓</td><td>FAD↓</td><td>CE↑</td><td>CU↑</td><td>PC↑</td><td>PQ↑</td><td>MUS↑</td><td>QLT↑</td></tr><tr><td rowspan="3">✓</td><td>H+C+S</td><td>9.44</td><td>5.60</td><td>7.57</td><td>7.66</td><td>6.00</td><td>8.25</td><td>3.55</td><td>3.77</td></tr><tr><td>H+C</td><td>10.49</td><td>5.16</td><td>7.43</td><td>7.57</td><td>6.42</td><td>8.15</td><td>3.43</td><td>3.46</td></tr><tr><td>H</td><td>11.35</td><td>7.58</td><td>7.46</td><td>7.50</td><td>5.80</td><td>8.09</td><td>3.14</td><td>3.41</td></tr><tr><td>X</td><td>H+C</td><td>109.76</td><td>8.86</td><td>7.06</td><td>7.40</td><td>5.92</td><td>7.71</td><td>-</td><td>-</td></tr></table>
+
+![](images/11bbfdc094d4c0697eea181011801f4dc9e0b996044e657a6ec95c97a9995318.jpg)  
+Figure 2: Aesthetic scores and sketch token accuracy of SongBloom-tiny with various patch sizes.
+
+![](images/bf9840de88e8153ea80be2830af3cefbe4265317d33d40d195086f3870944ade.jpg)  
+Figure 3: Aesthetic scores of SongBloom-tiny with increasing diffusion steps.
+
+# 6.3 Effect of Hyper-Parameters
+
+Figure 2 illustrates the impact of various patch sizes, where  $"+"\infty$  denotes separating the generation of sketch and acoustic sequence. For speech-oriented autoregressive diffusion models [7, 19], a small patch size is essential, as larger patch sizes make the model difficult to converge. For SongBloom, since the sketch tokens have served as the CoT of hidden vectors and guide the diffusion directly, a small patch size is no longer the optimal choice. Although smaller patch sizes provide more acoustic information for sketch generation, thereby improving sketch token accuracy, they also hinder the fluency of acoustic latents during the diffusion stage, as the window for preceding contexts becomes smaller. Figure 3 illustrates the relationship between performance and diffusion steps during inference. The RTF increases proportionally with the number of diffusion steps, while near-optimal generation performance is achieved in as few as 10 steps, indicating that the inference process can be further accelerated.
+
+# 7 Discussions and Limitations
+
+In this paper, we propose SongBloom, a novel approach for full-song generation that produces expressive, high-quality songs and achieves state-of-the-art performance across multiple metrics. Nevertheless, several open challenges remain. First, the current sketch representation is derived from SSL models, which lack interpretability. Replacing these with some symbolic formats could enable more fine-grained control and user customization, which will be our future work. Second, we believe that some reinforcement learning-based techniques, such as DPO [47] or PPO [48], can be applied to SongBloom, which aligns the generation process with user preferences, thereby enabling outputs that better match human aesthetic judgments.
+
+We fully acknowledge the potential ethical risks associated with music generation models. We ensure that both our models and training data are strictly used for academic research purposes only. We respect the intellectual property rights of all original artists and content creators. Every effort has been made to avoid the use of copyrighted material without proper authorization.
+
+# 8 Acknowledgement
+
+This work is supported by National Natural Science Foundation of China (Grant No. 62401377 and No. 62271432), Shenzhen Science and Technology Program (Shenzhen Key Laboratory, Grant No. ZDSYS20230626091302006), Program for Guangdong Introducing Innovative and Entrepreneurial Teams, Grant No. 2023ZT10X044.
+
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+
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+# A Ablation Study on Different Sketch Choices
+
+Table 5 demonstrates our early efforts at decomposing the semantic information of sketches. We compare different sketch representations, including pitch, pitch + chromagram, and semantic embeddings. As shown in the table, simple pitch-based sketches offer limited alignment capability, resulting in high PER and suboptimal performance in other metrics. The inclusion of chroma features slightly improves alignment, but still falls short of fully capturing the rich semantics needed for coherent vocal generation. These results validate the importance of incorporating abstract, semantically meaningful information into the sketch stage, and lay the groundwork for future exploration of interpretable yet powerful sketch formats.
+
+<table><tr><td>Type</td><td>PER(%)↓</td><td>FAD↓</td><td>CE↑</td><td>CU↑</td><td>PC↑</td><td>PQ↑</td></tr><tr><td>no sketch</td><td>109.76</td><td>8.86</td><td>7.06</td><td>7.40</td><td>5.92</td><td>7.71</td></tr><tr><td>pitch</td><td>103.22</td><td>5.43</td><td>7.34</td><td>7.58</td><td>6.03</td><td>8.04</td></tr><tr><td>pitch + chroma</td><td>68.47</td><td>5.47</td><td>7.37</td><td>7.57</td><td>5.98</td><td>8.06</td></tr><tr><td>SSL embedding</td><td>9.44</td><td>5.60</td><td>7.55</td><td>7.66</td><td>6.00</td><td>8.25</td></tr></table>
+
+Table 5: Impact of sketch types on SongBloom's performance.
+
+# B Time Complexity of SongBloom Inference
+
+We analyze the inference time complexity of SongBloom compared to a decoupled two-stage model, assuming both have the same number of layers.
+
+Let  $L_{1}$  denote the number of layers in the language model stage, and  $L_{2}$  the number of layers in the diffusion stage. Let  $T$  be the total number of frames for both semantic and acoustic sequences (eg.  $30\mathrm{s}\times 25\mathrm{fps} = 600$ ),  $P$  the patch size,  $N = T / P$  the number of patches, and  $S$  the number of diffusion steps.
+
+Assuming key-value caching is used during inference, we analyze the leading-order time complexity:
+
+Decoupled 2-stage model:
+
+$$
+O _ {0} = L _ {1} \cdot \frac {(1 + T) T}{2} + L _ {2} \cdot (2 T) ^ {2} \cdot S
+$$
+
+SongBloom:
+
+$$
+O _ {1} = L _ {1} \cdot \frac {(1 + T + N) (T + N)}{2} + L _ {2} \cdot (2 P) ^ {2} \cdot N \cdot S
+$$
+
+Substituting  $N = T / P$ , we compute the difference:
+
+$$
+O _ {1} - O _ {0} = \left(\frac {L _ {1}}{2 P ^ {2}} + \frac {L _ {1}}{P} - 4 L _ {2} S\right) T ^ {2} + \left(\frac {L _ {1}}{2 P} + 4 L _ {2} P S\right) T
+$$
+
+When  $T$  is sufficiently large, the  $T^2$  term dominates. In most practical cases, where:
+
+$$
+L _ {1} <   4 S P \cdot \frac {2 P}{2 P + 1} \cdot L _ {2}
+$$
+
+the coefficient of  $T^2$  is negative. Therefore, we conclude that SongBloom is asymptotically more efficient than the decoupled two-stage models, owing to its patch-wise diffusion mechanism and reduced per-step input length during inference.
+
+# C Criteria of the Subjective Listening Test
+
+1. Musicality of vocal: (1-5 points) Does the main melody of the generated vocal match the subjective expectation?
+
+- 5 points: The melody is pleasant and emotionally expressive, with strong musical phrasing. It aligns well with expectations.  
+- 4 points: The melody generally meets expectations and conveys the song's theme and emotion, but lacks standout features.  
+- 3 points: The melody mostly aligns with expectations and conveys the theme and emotion, though some notes feel abrupt.  
+- 2 points: Only parts of the melody are coherent; most notes are scattered, and the theme and emotion are vaguely presented.  
+- 1 point: The melody significantly deviates from expectations, lacks coherent musical phrasing, and fails to convey the song's theme and emotion.
+
+2. Musicality of accompaniment: (1-5 points) Does the accompaniment of the generated song sound harmonious?
+
+- 5 points: The accompaniment is richly colored and features diverse instrumentation. The melody is beautiful and complements the main melody harmoniously.  
+- 4 points: The accompaniment supports the main melody, but uses limited instrumentation or has a generally average melodic performance.  
+- 3 points: The accompaniment mostly supports the main melody, with only minor discord. However, it sometimes clashes with the main melody and lacks variety and color in instrumentation.  
+- 2 points: Some segments show disorganized instrumentation and monotonous melody, barely supporting the main melody.  
+- 1 point: The instrumentation is chaotic and the melody is discordant. There is a clear conflict with the main melody, failing to provide support.
+
+3. Quality of vocal: (1-5 points) Is the vocal in the generated music clear and bright, with a full high-frequency range? Are there any noises or distortions present?
+
+- 5 points: The vocal quality is rich and clear, with no noise, approaching studio-recording quality.  
+- 4 points: The vocal quality is relatively clear, with slight noise that is either imperceptible or barely noticeable.  
+- 3 points: The vocal quality contains some noise or distortion, but it does not significantly affect the listening experience.  
+- 2 points: The vocal quality is unclear and unstable, resulting in a poor listening experience. Noticeable noise or distortion is present.  
+- 1 point: The vocal quality is extremely poor, with an unpleasant listening experience, and the vocal characteristics are barely recognizable.
+
+4. Quality of accompaniment: (1-5 points) Is the high-frequency range of the generated music's accompaniment full? Are there any noises or instrumental distortions?
+
+- 5 points: The accompaniment has a full and clear sound quality with no flaws. The characteristics and melodies of different instruments are clearly distinguishable.  
+- 4 points: The accompaniment has good sound quality with slight noise. Only a few instruments in certain segments are hard to distinguish or slightly distorted, but this does not affect the overall listening experience.  
+- 3 points: The accompaniment has average sound quality. Some instruments are unclear or unidentifiable in certain segments. There is noticeable noise, distortion, or a lack of clarity.  
+- 2 points: The accompaniment has poor sound quality. In most parts of the piece, most instruments are unrecognizable. There is clear noise, distortion, or lack of clarity.  
+- 1 point: The accompaniment has extremely poor sound quality, with severe distortion, making it nearly impossible to identify any instrumental characteristics.
+
+5. Correctness of lyrics: (1-4 points) Does the song content match the lyrics? Are there any errors such as extra words, missing words, or mechanical repetition?
+
+- 4 points: The song content fully matches the lyrics, with no missing or extra words, and no mechanical repetition of musical segments.
+
+- 3 points: The generated song contains a small number (within 5 words) of unclear, repeated, or missing lyrics.  
+- 2 points: The generated song contains multiple segments with unclear, repeated, or missing lyrics.  
+- 1 point: The generated song does not match the lyrics at all.
+
+6. Consistency of prompt: (1-5 points) Does the musical style of the generated song match the style of the reference audio prompt?
+
+- 5 points: The musical style of the generated song fully matches the style specified in the prompt.  
+- 4 points: The musical style of the generated song is similar to the specified prompt, with only slight differences in some segments.  
+- 3 points: The musical style is somewhat similar to the specified style, but only vaguely reflects its characteristics.  
+- 2 points: The musical style does not resemble the specified style, with only faint traces of the intended musical elements.  
+- 1 point: The musical style has no relation to the specified style, making it difficult to connect the prompt with the resulting music.

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