| Notation | Description |
| x | Time series represented as a bold lowercase symbol |
| F(x) | Discrete Fourier transformation of time series x |
| ∠ | The complex argument for obtaining phase values |
| ω | Variable that represents the frequencies in radian |
| |X(ejω)| | Magnitude components of the Fourier transformation of a time series |
| φ(ω) | Phase components of the Fourier transformation of a time series |
| ω0 | Frequency of the specific harmonic whose period is equal or longer than the sample x |
| T0 = 2π/ω0 | Period of the specific harmonic with frequency ω0 |
| Φ | Random variable for phase angles, i.e., φ ~ Φ |
| φ(ω0) | Phase angle of a specific harmonic with the frequency ω0 |
| φx(t) | Variable that represents phase angles of all harmonics for sample x(t) |
| φx(t-t') | Variable that represents phase angles for sample x(t-t') |
| φx(t)(ω0) | Phase angle of the specific harmonic with the frequency ω0 for sample x(t) |
| T(x,φ) | The proposed transformation function |
| Δφ | Phase difference between two angles |
| Δφx(t) | Phase difference between the given angle φ in the transformation and the harmonic with frequency ω0 when sample x(t) is decomposed using Fourier transformation |
| Δφx(t-t') | Phase difference between the given angle φ in the transformation and the harmonic with frequency ω0 when sample x(t-t') is decomposed using Fourier transformation |
| Mφ | Manifold notation with an angle φ |
| Mφa | A specific manifold with the angle φa |
| T(x,φa) | The output of the transformation, i.e., a time series that lies on the manifold Mφa |
| F(T(x,φ)) | Fourier transformation of the output from the proposed transformation: Since the transformation function produces a tuple, we specifically apply the Fourier transformation to the first output, which corresponds to the time series. |
| fθ(.) | Parametric mapping with parameter θ |
| fGθ: Rd→Φ | The guidance network that outputs an angle to map the sample to a specific manifold |
| fCθ: X→Y | The classifier neural network |
| Var(.) | Variance function |
| % | Modulo operation |
+
+Table 8: Detailed list of notations used in this work
+
+# A.3 PROOFS
+
+Lemma A.1 (Circular Shift). Given a sample $\mathbf{x}$ in the interval $[0, t_{int}]$ and its shifted version $\mathbf{x}(t - t')$ , where the shift is a random value from the finite real numbers, i.e., $t' \in (-\infty, \infty)$ . The shift is periodic with the signal length $t_{int}$ , leading to the same vector representation when the shift is an integer multiple of the signal length.
+
+$$
+\infty > t > | t ^ {\prime} | > - \infty , 0 = t ^ {\prime} (\mathrm {m o d} t _ {i n t}) \Longrightarrow \mathbf {x} (t) = \mathbf {x} (t - t ^ {\prime})
+$$
+
+Proof. From circular shift, we know
+
+$$
+\mathbf {x} (t) = \mathbf {x} (t (\mathrm {m o d} t _ {\text {i n t}})) \tag {15}
+$$
+
+$$
+\mathbf {x} \left(t - t ^ {\prime}\right) = \mathbf {x} \left(\left(t - t ^ {\prime}\right) (\mathrm {m o d} t _ {i n t})\right) \tag {16}
+$$
+
+Therefore, $0 = t'$ (mod $t_{int}$ ) $\Longrightarrow \mathbf{x}(t) = \mathbf{x}(t - t')$
+
+Throughout the proofs, the length of the samples and their intervals are denoted by the same variable $t$ . In other words, the samples are assumed to start at $t = 0$ and finish at $t_{int} = t$ .
+
+# A.4 PROOF FOR PROPOSITION 2.1
+
+Proposition A.2 (Time shift as a Group Operation). Shift operation in time domain defines an Abelian Group of phase angles in the frequency domain for each harmonic with frequency $\omega_{k}$
+
+$$
+\left(\Phi_ {k}, + \mod 2 \pi\right), \text {w h e r e} \Phi_ {k} = \phi \mid \phi = \left(\phi \left(\omega_ {k}\right) + \omega_ {k} t ^ {\prime}\right) \mod 2 \pi , t ^ {\prime} \in \mathbb {R} \tag {17}
+$$
+
+Proof. Let $\mathbf{x}(t + t^{\prime})$ be randomly shifted variant of sample $\mathbf{x}(t)$ where $t^\prime \in \mathbb{R}$ . We can decompose these two sequences as in below using Fourier transformation.
+
+$$
+\mathbf {x} (t) \xrightarrow {\mathcal {F} (\cdot)} | X (e ^ {j \omega}) | e ^ {j \phi (\omega)} \text {a n d} \mathbf {x} (t + t ^ {\prime}) \xrightarrow {\mathcal {F} (\cdot)} | X (e ^ {j \omega}) | e ^ {j (\phi (\omega) + \omega t ^ {\prime})} \tag {18}
+$$
+
+The phase values of these two time series for a harmonic at the given frequency can be expressed as follows.
+
+$$
+\phi_ {\mathbf {x} (t)} \left(\omega_ {k}\right) = \phi \left(\omega_ {k}\right) \text {a n d} \phi_ {\mathbf {x} \left(t - t ^ {\prime}\right)} \left(\omega_ {k}\right) = \phi \left(\omega_ {k}\right) + \omega_ {k} t ^ {\prime} \tag {19}
+$$
+
+Then, we can define a set of phase angles $\varPhi_{k}$ with shift values $t^\prime$
+
+$$
+\Phi_ {k} = \left\{\phi \mid \phi = \left(\phi \left(\omega_ {k}\right) + \omega_ {k} t ^ {\prime}\right) \bmod 2 \pi , t ^ {\prime} \in \mathbb {R} \right\} \tag {20}
+$$
+
+This set of phase angles with time shift operation defines the circle group $\mathbb{T}$ . The circle group is Abelian (Fuchs, 1960), with time shifts corresponding to multiplication in the complex plane. For completeness, we have shown all the group axioms.
+
+# GROUP AXIOMS
+
+The set of phase angles with time shift, $\varPhi_{k}=\{\phi\mid\phi=(\phi(\omega_{k})+\omega_{k}t^{\prime})\mod 2\pi,t^{\prime}\in\mathbb{R}\}$ , satisfies the five axioms of an Abelian group under modular addition.
+
+# Axiom 1: Closure
+
+For any two phase angles $\phi_1, \phi_2 \in \varPhi_k$ , their sum is also in $\varPhi_k$ .
+
+Let $\phi_1 = (\phi(\omega_k) + \omega_k t_1') \mod 2\pi$ and $\phi_2 = (\phi(\omega_k) + \omega_k t_2') \mod 2\pi$ . Then their sum is:
+
+$$
+\phi_ {1} + \phi_ {2} = \left(\phi \left(\omega_ {k}\right) + \omega_ {k} t _ {1} ^ {\prime} + \phi \left(\omega_ {k}\right) + \omega_ {k} t _ {2} ^ {\prime}\right) \mod 2 \pi
+$$
+
+$$
+\phi_ {1} + \phi_ {2} = \left(\phi \left(\omega_ {k}\right) + \omega_ {k} \left(t _ {1} ^ {\prime} + t _ {2} ^ {\prime}\right)\right) \mod 2 \pi
+$$
+
+Since $t_1' + t_2' \in \mathbb{R}$ , the sum is also in $\varPhi_k$ , so closure holds.
+
+# Axiom 2: Associativity
+
+For any three phase angles $\phi_1, \phi_2, \phi_3 \in \varPhi_k$ , their sum is associative under addition modulo $2\pi$ .
+
+$$
+\left(\left(\phi_ {1} + \phi_ {2}\right) \mod 2 \pi\right) + \phi_ {3}) \mod 2 \pi = \left(\phi_ {1} + \left(\phi_ {2} + \phi_ {3}\right) \mod 2 \pi\right) \mod 2 \pi
+$$
+
+Let $\phi_1 = (\phi(\omega_k) + \omega_k t_1') \mod 2\pi$ , $\phi_2 = (\phi(\omega_k) + \omega_k t_2') \mod 2\pi$ , and $\phi_3 = (\phi(\omega_k) + \omega_k t_3') \mod 2\pi$ . Then:
+
+$$
+\begin{array}{l} \left(\left(\phi_ {1} + \phi_ {2}\right) \mod 2 \pi\right) + \phi_ {3} = \left(\phi (\omega_ {k}) + \omega_ {k} t _ {1} ^ {\prime} + \omega_ {k} t _ {2} ^ {\prime}\right) \mod 2 \pi + \phi_ {3} \\ = \left(\phi (\omega_ {k}) + \omega_ {k} (t _ {1} ^ {\prime} + t _ {2} ^ {\prime}) \mod 2 \pi + \omega_ {k} t _ {3} ^ {\prime}\right) \mod 2 \pi \\ \end{array}
+$$
+
+Similarly, for the right-hand side:
+
+$$
+\begin{array}{l} \phi_ {1} + \left(\left(\phi_ {2} + \phi_ {3}\right) \mod 2 \pi\right) = \phi_ {1} + \left(\phi (\omega_ {k}) + \omega_ {k} \left(t _ {2} ^ {\prime} + t _ {3} ^ {\prime}\right) \mod 2 \pi\right) \\ = \left(\phi \left(\omega_ {k}\right) + \omega_ {k} t _ {1} ^ {\prime} + \omega_ {k} \left(t _ {2} ^ {\prime} + t _ {3} ^ {\prime}\right)\right) \mod 2 \pi \\ \end{array}
+$$
+
+Using $(a + b) \mod 2\pi = ((a \mod 2\pi) + (b \mod 2\pi)) \mod 2\pi$ , both sides simplify to $(\phi(\omega_k) + \omega_k(t_1' + t_2' + t_3')) \mod 2\pi$ . Thus, associativity holds under addition modulo $2\pi$ .
+
+Axiom 3: Identity Element
+
+The identity element in this group is the phase angle when no time shift has occurred, i.e., $t' = 0$ .
+
+$$
+\phi_ {0} = \left(\phi \left(\omega_ {k}\right) + \omega_ {k} \cdot 0\right) \mod 2 \pi = \phi \left(\omega_ {k}\right)
+$$
+
+For any $\phi_1\in \varPhi_k$ , we have:
+
+$$
+\phi_ {1} + \phi_ {0} = \phi_ {1}
+$$
+
+Thus, $\phi_0$ is the identity element.
+
+Axiom 4: Inverse Element
+
+For any phase angle $\phi_{1} = (\phi (\omega_{k}) + \omega_{k}t^{\prime})$ mod $2\pi$ , its inverse is:
+
+$$
+\phi_ {1} ^ {- 1} = \left(- \omega_ {k} t ^ {\prime}\right) \mod 2 \pi
+$$
+
+Then:
+
+$$
+\phi_ {1} + \phi_ {1} ^ {- 1} = \left(\phi \left(\omega_ {k}\right) + \omega_ {k} t ^ {\prime} - \omega_ {k} t ^ {\prime}\right) \mod 2 \pi = \phi \left(\omega_ {k}\right)
+$$
+
+Thus, each element has an inverse.
+
+Axiom 5: Commutativity
+
+For any two phase angles $\phi_1, \phi_2 \in \varPhi_k$ , the sum is commutative:
+
+$$
+\phi_ {1} + \phi_ {2} = \phi_ {2} + \phi_ {1}
+$$
+
+This follows from the commutativity of modular addition, so the group is Abelian.
+
+Proposition 2.1 states that shift operation in time domain defines an Abelian group of phase angles for each harmonic. Proposition 2.1 holds a key role in our algorithm, as it establishes an abstract connection between the time-domain shift operation and its effects on samples in the frequency domain.
+
+# A.4.1 PROOF FOR THEOREM 2.2
+
+Theorem A.3 (Covering the Entire Time Space Injectively). Given a sample $\mathbf{x}$ , the defined function $\mathcal{T}(\mathbf{x},\phi):\varPhi\times\mathbb{R}^d\to\mathbb{R}^d\times\varDelta\varPhi$ is bijective such that all shift variants of a sample can be covered with the unique phase angle of a harmonic whose period is longer or equal to the length of $\mathbf{x}$ .
+
+$$
+\forall \phi_ {a}, \phi_ {b} \in \varPhi , \mathcal {T} (\mathbf {x}, \phi_ {a}) = \mathcal {T} (\mathbf {x}, \phi_ {b}) \Longrightarrow \phi_ {a} = \phi_ {b}
+$$
+
+$$
+\forall t ^ {\prime} \in \mathbb {R}, \exists \phi \in \Phi , \mathcal {T} (\mathbf {x}, \phi) = (\mathbf {x} (t - t ^ {\prime}), \Delta \phi),
+$$
+
+Proof.
+
+$$
+\mathbf {x} (t) \xrightarrow {\mathcal {F} (\cdot)} | X (e ^ {j \omega}) | e ^ {j \phi (\omega)} \tag {21}
+$$
+
+$$
+\mathbf {x} \left(t - t ^ {\prime}\right) \xrightarrow {\mathcal {F} (.)} | X \left(e ^ {j \omega}\right) | e ^ {j (\phi (\omega) - \omega t ^ {\prime})} \tag {22}
+$$
+
+$$
+\phi_ {\mathbf {x} (t)} = \phi (\omega), \quad \phi_ {\mathbf {x} (t - t ^ {\prime})} = \phi (\omega) - \omega t ^ {\prime} \tag {23}
+$$
+
+$$
+\phi_ {\mathbf {x} (t)} - \phi_ {\mathbf {x} (t - t ^ {\prime})} = \omega t ^ {\prime}, \tag {24}
+$$
+
+Using Euler's formula,
+
+$$
+\phi_ {\mathbf {x} (t)} - \phi_ {\mathbf {x} (t - t ^ {\prime})} = (\omega t ^ {\prime}) (\mathrm {m o d} 2 \pi) \tag {25}
+$$
+
+$$
+\phi_ {\mathbf {x} (t)} - \phi_ {\mathbf {x} (t - t ^ {\prime})} = \left(\frac {2 \pi}{T} t ^ {\prime}\right) (\mathrm {m o d} 2 \pi) \tag {26}
+$$
+
+$$
+\forall T _ {0} \in T, \infty > T _ {0} \geq t \Rightarrow \phi_ {\mathbf {x} (t)} (\omega_ {0}) - \phi_ {\mathbf {x} (t - t ^ {\prime})} (\omega_ {0}) = \frac {2 \pi}{T _ {0}} t ^ {\prime} \tag {27}
+$$
+
+Thus,
+
+$$
+\phi_ {\mathbf {x} (t)} \left(\omega_ {0}\right) - \phi_ {\mathbf {x} \left(t - t ^ {\prime}\right)} \left(\omega_ {0}\right) = \phi_ {\mathbf {x} (t)} \left(\omega_ {0}\right) - \phi_ {\mathbf {x} \left(t - t ^ {\prime}\right)} \left(\omega_ {0}\right) \Longrightarrow t ^ {\prime} = t ^ {\prime} \tag {28}
+$$
+
+Also,
+
+$$
+\forall T _ {0} \in T, \infty > t > T _ {0} \Rightarrow \phi_ {\mathbf {x} (t)} (\omega_ {0}) - \phi_ {\mathbf {x} (t - t ^ {\prime})} (\omega_ {0}) = \left(\frac {2 \pi}{T _ {0}} t ^ {\prime}\right) \mod 2 \pi , \tag {29}
+$$
+
+$$
+\because \exists t ^ {\prime} \in \mathbb {R}, \left(\frac {2 \pi}{T _ {0}} t ^ {\prime}\right) > 2 \pi \tag {30}
+$$
+
+$$
+\therefore \exists t ^ {\prime}, t ^ {\prime} \in \mathbb {R}, \neg \left(\phi_ {\mathbf {x} (t)} \left(\omega_ {0}\right) - \phi_ {\mathbf {x} \left(t - t ^ {\prime}\right)} \left(\omega_ {0}\right) = \phi_ {\mathbf {x} (t)} \left(\omega_ {0}\right) - \phi_ {\mathbf {x} \left(t - t ^ {\prime}\right)} \left(\omega_ {0}\right) \Longrightarrow t ^ {\prime} = t ^ {\prime}\right), \tag {31}
+$$
+
+which proves injection. Similarly, for surjection using Lemma A.1,
+
+$$
+\forall t ^ {\prime} \in \mathbb {R}, \infty > T _ {0} \geq t > | t ^ {\prime} | \Rightarrow \phi_ {\mathbf {x} (t)} (\omega_ {0}) - \phi_ {\mathbf {x} (t - t ^ {\prime})} (\omega_ {0}) = \frac {2 \pi}{T _ {0}} t ^ {\prime} \tag {32}
+$$
+
+
+
+The final equation completes the proof by showing that the phase difference between the sample and its shifted version can get unique values for any shift, i.e., covering the whole time space.
+
+# A.4.2 PROOF FOR THEOREM 2.3
+
+Theorem A.4 (Guarantees for Shift-Invariancy). Given a sample $\mathbf{x}$ and a randomly shifted variant of it $\mathbf{x}(t - t')$ , if the transformation function $\mathcal{T}(\mathbf{x},\phi)$ is applied to both samples with the same angle $\phi_{a}$ , the resulting time series will be the same while carrying the same information.
+
+$$
+\mathcal {T} (\mathbf {x} (t), \phi_ {a}) = \left(\tilde {\mathbf {x}} (t), \varDelta \phi_ {\mathbf {x} (t)}\right), \quad \mathcal {T} (\mathbf {x} (t - t ^ {\prime}), \phi_ {a}) = \left(\tilde {\mathbf {x}} (t), \varDelta \phi_ {\mathbf {x} (t - t ^ {\prime})}\right)
+$$
+
+Proof.
+
+$$
+\mathbf {x} (t) \xrightarrow {\mathcal {F} (\cdot)} | X (e ^ {j \omega}) | e ^ {j \phi (\omega)}, \quad \mathbf {x} (t - t ^ {\prime}) \xrightarrow {\mathcal {F} (\cdot)} | X (e ^ {j \omega}) | e ^ {j \phi (\omega)} e ^ {\omega t ^ {\prime}} \tag {33}
+$$
+
+$$
+\phi_ {\mathbf {x} (t)} = \phi (\omega), \quad \phi_ {\mathbf {x} (t - t ^ {\prime})} = \phi (\omega) - \omega t ^ {\prime} \tag {34}
+$$
+
+$$
+\phi_ {\mathbf {x} (t - t ^ {\prime})} - \phi_ {\mathbf {x} (t)} = - \omega t ^ {\prime} \tag {35}
+$$
+
+Using Equation 11, the phase difference between samples $(\mathbf{x}(t),\mathbf{x}(t - t'))$ and $\phi_{a}$ can be obtained as,
+
+$$
+\Delta \phi_ {\mathbf {x} (t)} = \frac {[ \phi (\omega_ {0}) - \phi_ {a} ]}{2 \pi} \cdot T _ {0}, \quad \Delta \phi_ {\mathbf {x} (t - t ^ {\prime})} = \frac {[ \phi (\omega_ {0}) - \omega_ {0} t ^ {\prime} - \phi_ {a} ]}{2 \pi} \cdot T _ {0} \tag {36}
+$$
+
+$$
+\Delta \phi_ {\mathbf {x} (t - t ^ {\prime})} - \Delta \phi_ {\mathbf {x} (t)} = - \omega_ {0} t ^ {\prime} \frac {T _ {0}}{2 \pi} \tag {37}
+$$
+
+Using Fourier transform as in Equation 4,
+
+$$
+\mathcal {F} \left\{\mathcal {T} (\mathbf {x} (t), \phi_ {a}) \right\} = | X (e ^ {j \omega}) | e ^ {j \phi (\omega)} e ^ {- j \omega \Delta \phi_ {\mathbf {x}} (t)} \tag {38}
+$$
+
+$$
+\mathcal {F} \left\{\mathcal {T} \left(\mathbf {x} \left(t - t ^ {\prime}\right), \phi_ {a}\right) \right\} = | X \left(e ^ {j \omega}\right) | e ^ {j \phi (\omega)} e ^ {- j \omega t ^ {\prime}} e ^ {- j \omega \Delta \phi_ {\mathbf {x} _ {(t - t ^ {\prime})}}} \tag {39}
+$$
+
+Given that the amplitudes are identical, demonstrating equality in phase is sufficient, as shown below,
+
+$$
+\phi_ {\mathcal {T} (\mathbf {x} (t), \phi_ {a})} = \phi (\omega) - \omega \Delta \phi_ {\mathbf {x} (t)}, \quad \phi_ {\mathcal {T} (\mathbf {x} (t - t ^ {\prime}), \phi_ {a})} = \phi (\omega) - \omega t ^ {\prime} - \omega \Delta \phi_ {\mathbf {x} _ {(t - t ^ {\prime})}} \tag {40}
+$$
+
+$$
+\phi_ {\mathcal {T} (\mathbf {x} (t - t ^ {\prime}), \phi_ {a})} - \phi_ {\mathcal {T} (\mathbf {x} (t), \phi_ {a})} = - \omega t ^ {\prime} - \omega \Delta \phi_ {\mathbf {x} _ {(t - t ^ {\prime})}} + \omega \Delta \phi_ {\mathbf {x} (t)} \tag {41}
+$$
+
+$$
+\phi_ {\mathcal {T} (\mathbf {x} (t - t ^ {\prime}), \phi_ {a})} - \phi_ {\mathcal {T} (\mathbf {x} (t), \phi_ {a})} = - \omega t ^ {\prime} - \omega [ \Delta \phi_ {\mathbf {x} _ {(t - t ^ {\prime})}} - \Delta \phi_ {\mathbf {x} (t)} ] \tag {42}
+$$
+
+$$
+\phi_ {\mathcal {T} (\mathbf {x} (t - t ^ {\prime}), \phi_ {a})} - \phi_ {\mathcal {T} (\mathbf {x} (t), \phi_ {a})} = - \omega t ^ {\prime} - \omega \left[ - \omega_ {0} t ^ {\prime} \frac {T _ {0}}{2 \pi} \right] \tag {43}
+$$
+
+$$
+\phi_ {\mathcal {T} (\mathbf {x} (t - t ^ {\prime}), \phi_ {a})} - \phi_ {\mathcal {T} (\mathbf {x} (t), \phi_ {a})} = - \omega t ^ {\prime} + \omega \left[ \frac {2 \pi}{T _ {0}} t ^ {\prime} \frac {T _ {0}}{2 \pi} \right] \tag {44}
+$$
+
+$$
+\phi_ {\mathcal {T} (\mathbf {x} (t - t ^ {\prime}), \phi_ {a})} - \phi_ {\mathcal {T} (\mathbf {x} (t), \phi_ {a})} = - \omega t ^ {\prime} + \omega t ^ {\prime} \tag {45}
+$$
+
+$$
+\phi_ {\mathcal {T} (\mathbf {x} (t - t ^ {\prime}), \phi_ {a})} = \phi_ {\mathcal {T} (\mathbf {x} (t), \phi_ {a})} \tag {46}
+$$
+
+$$
+\mathcal {F} \left\{\mathcal {T} (\mathbf {x} (t), \phi_ {a}) \right\} = | X \left(e ^ {j \omega}\right) \left| e ^ {j \phi_ {\mathcal {T}} (\mathbf {x} (t), \phi_ {a})} \right. \tag {47}
+$$
+
+$$
+\mathcal {F} \left\{\mathcal {T} \left(\mathbf {x} \left(t - t ^ {\prime}\right), \phi_ {a}\right) \right\} = | X \left(e ^ {j \omega}\right) | e ^ {j \phi_ {\mathcal {T}} \left(\mathbf {x} \left(t - t ^ {\prime}\right), \phi_ {a}\right)} \tag {48}
+$$
+
+Therefore, the output time series samples will be the same after applying the transformation.
+
+We complete the proof by showing the phase and magnitude of Fourier transformation of both samples are the same after transformation even though a random unknown shift is applied to the sample. Thus, the proposed transformation guarantees shift-invariancy without limiting the range of shifts.
+
+# B ALGORITHM
+
+In this section, we present the pseudocode and the PyTorch (Paszke et al, 2019) implementation for the proposed transformation function. Algorithm 1 details each step of the transformation, which takes a sample, $\mathbf{x}$ , and an angle, $\phi$ as inputs and outputs the transformed sample.
+
+1: Input: $\mathbf{x}$ , $\phi_{a}$
+2: Output: $\mathcal{T}(\mathbf{x},\phi_a)$
+
+3: $|X(e^{j\omega})|e^{j\phi(\omega)} = \int_{-\infty}^{\infty} \mathbf{x}(t)e^{-j\omega t}$
+
+$\triangleright$ Calculate the Fourier transformation to obtain harmonics
+
+4: $\phi (\omega_0) = \angle \left(|X(e^{j\omega_0})|e^{j\phi (\omega_0)}\right)$
+
+Obtain the angle for the harmonic with period $\mathrm{T_0}$
+
+5: $\theta = [\phi (\omega_0) - \phi_a]\% 2\pi$
+
+6: $\Delta \phi = \left\{ \begin{array}{ll} \frac{(\theta - 2\pi)*T_0}{2\pi}, & \text{if } \theta > \pi \\ \frac{\theta*T_0}{2\pi}, & \text{else} \end{array} \right.$
+
+$\triangleright$ Calculate the phase difference between the harmonic and the angle $\phi$
+
+7: $|X(e^{j\omega})|e^{j(\phi (\omega) - \omega \Delta \phi)} = |X(e^{j\omega})|e^{j\phi (\omega)}*e^{-j\omega \Delta \phi}$
+
+Apply a linear phase shift to each harmonic
+
+8: Return: $\mathcal{F}^{-1}(|X(e^{j\omega})|e^{j(\phi (\omega) - \omega \Delta \phi)})$
+
+Below, we provide the PyTorch implementation of our proposed transformation function, which includes two functions. The first function, distanceCalculate, computes the phase difference between the harmonic with frequency $\omega_0$ and the desired input angles. The second function, diffeomorphism, performs the main transformation: it takes as inputs the batch of samples $\mathbf{x}$ and the angles $\phi$ , and outputs the transformed samples in the time domain.
+
+Algorithm 1 Algorithm for the proposed diffeomorphism.
+```python
+def distanceCalculate(angleDiff):
+ theta = angleDiff % (2 * torch.pi)
+ # Calculate the angular distance on the unit circle
+ theta[theta > torch.pi] == 2 * torch.pi
+ return theta
+def diffeomorphism(sample, desiredAngles):
+ B, L, D = sample.shape
+ samplesFFT = torch fft.rfft(sample, dim=1)
+ freq = torch fft.rfftfreq(n=L)
+ phAngle = torch.angle(sampleFFT)
+ # Get the phase angle of the harmonic with frequency T_0
+ angles = phAngle[torch/arange(phAngle.size(0)), 1, 0].squeeze()
+ # Calculate the angle difference
+ theta = distanceCalculateAngles-desiredAngles)
+ #Normalize it to the sepecific harmonic with frequency w_0
+ dtheta = theta / (2*torch.pi*freq[1])
+ #Create complex exponentials with specific phase values
+ linShift = torch.exp(-1j*2*torch.pi*freq[None,:]*dtheta[:, None])
+ linShift = linShift.unsqueeze(dim=2).expand(-1, -1, D)
+ #Apply a linear phase shift to all harmonics
+ shiftedFFT = linShift*(samplesFFT)
+ #Return to the time domain
+ transformedSamples = torch.fft.irfftshiftedFFT, n=L, dim=1)
+ return transformedSamples
+```
+
+Implementation 1: PyTorch implementation of the proposed transformation
+
+# C EXPERIMENTS
+
+Here, we give a detailed description of datasets, architectures, metrics, and training details for our experiments. We performed our experiments on NVIDIA GeForce RTX 4090 GPUs, involving training with three random seeds for all datasets, totaling approximately 480 GPU hours including ablation. We reported the mean of three runs with the standard deviation.
+
+# C.1 DATASETS
+
+In this section, we give details about the datasets that are used during our experiments. Overall, we have used eight datasets with six different time series tasks including heart rate prediction, cardiovascular disease classification, activity recognition, step counting, sleep stage classification, and lung sound classification from six sensor modalities photoplethysmography, electrocardiogram, inertial measurement units, electroencephalography, and audio. When selecting datasets for our experiments, we prioritized signals that provide meaningful insights into individuals' mental and physical health, where robust inference is particularly critical. Therefore, we specifically choose signals generated by humans. Additional to main experiments, in appendix, we also included lung audio classification from audio signals.
+
+# C.1.1 HEART RATE PREDICTION
+
+IEEE SPC The IEEE SPC dataset overall has 22 recordings of 22 subjects, ages ranging from 18 to 58 performing three different activities (Rocha et al., 2020). Each recording has sampled data from three accelerometer signals and two PPG signals along with the ECG data with a sampling frequency of $125\mathrm{Hz}$ . All these recordings were recorded from the wearable device placed on the wrist of each individual. All recordings were captured with two 2-channel PPGs with green LEDs, a tri-axial accelerometer, and a chest ECG for the ground-truth HR estimation. We averaged the two channels of PPG for prediction. We choose the last five subjects of SPC22 to be used for source domains. Throughout our experiments, we used PPG channels without integrating any inertial measurements.
+
+Dalia The PPG dataset for motion compensation and heart rate estimation in Daily Life Activities (DaLiA) was recorded from 15 subjects (8 females, 7 males, mean age of 30.6), where each recording was approximately two hours long. PPG signals were recorded while subjects went through different daily life activities, for instance sitting, walking, driving, cycling, working, and so on. PPG signals were recorded at a sampling rate of $64\mathrm{Hz}$ . The first five subjects are used as source domains, similar to Demirel & Holz (2023).
+
+We standardize all PPG datasets as follows, same as the previous works (Biswas et al., 2019). Initially, a fourth-order Butterworth bandpass filter with a frequency range of $0.5 - 4\mathrm{Hz}$ is applied to PPG signals. Subsequently, a sliding window of 8 seconds with 2-second shifts is employed for segmentation, followed by z-score normalization of each segment. Lastly, the signal is resampled to a frequency of $25\mathrm{Hz}$ for each segment. We used an 8-block ResNet model with a stride of 2, a learning rate of $5e - 4$ and a batch size of 32. Additional results with further experiments can be found in Appendix D.
+
+# C.1.2 HUMAN ACTIVITY RECOGNITION
+
+UCIHAR Human activity recognition using a smartphone's dataset (UCIHAR) (Anguita et al., 2012) is collected by 30 subjects within the age range of 16 to 48 performing six daily living activities with a waist-mounted smartphone. Six activities include walking, sitting, lying, standing, walking upstairs, and walking downstairs. Data is captured by 3-axial linear acceleration and 3-axial angular velocity at a constant rate of $50\mathrm{Hz}$ . We used the pre-processing technique the same as in (Qian et al., 2021) such that the input contains nine channels with 128 features (it is sampled in a sliding window of 2.56 seconds and $50\%$ overlap, resulting in 128 features for each window). Windows are normalized to a mean of zero and unit standard deviation before feeding it to the models. The experiments are conducted with a leave-one-domain-out strategy with the first five subjects, where one of the domains is chosen to be the unseen target (Qian et al., 2022). We used an 8-block ResNet model with a stride of 2, a learning rate of $3e - 3$ , and a batch size of 32.
+
+HHAR Heterogeneity Dataset for Human Activity Recognition (HHAR) is collected by nine subjects within an age range of 25 to 30 performing six daily living activities with eight different smartphones—Although HHAR includes data from smartwatches as well, we use data from smartphones—that were kept in a tight pouch and carried by the users around their waists (Stisen et al., 2015). Subjects then perform six activities including cycling, sitting, descending stairs, ascending stairs, standing, and walking. Considering the variable sampling frequencies of smart devices in the HHAR dataset, we downsampled the readings to $50\mathrm{Hz}$ . We employed sliding windows with lengths of 100 (two seconds) and 50, using a specified step size. These windows were then normalized to a mean of zero with unit standard deviation. In our experiments, we utilized the data from the first four subjects (i.e., a, b, c, d) as source domains, following a similar approach to previous papers (Qian et al., 2022; Demirel & Holz, 2023). We used an 8-block ResNet model with a stride of 2, a learning rate of $1e-3$ , and a batch size of 64. The learning rate $1e-3$ for the guidance network, was the same for the activity recognition task.
+
+# C.1.3 CARDIOVASCULAR DISEASE (CVD) CLASSIFICATION
+
+Chapman Chapman University, Shaoxing People's Hospital (Chapman) ECG dataset which provides 12-lead ECG with a 10-second sampling rate of $500\mathrm{Hz}$ . The recordings are downsampled to $100\mathrm{Hz}$ , resulting in each ECG frame consisting of 1000 samples. The labeling setup follows the same approach as in Zheng et al. (2020) with four classes: atrial fibrillation, GSVT, sudden bradycardia, and sinus rhythm. The ECG frames are normalized to have a mean of 0 and scaled to have a standard deviation of 1. We split the dataset to $80 - 20\%$ for training and testing as suggested in Zheng et al. (2020). We chose leads I, II, III, and V2 during our experiments for both ECG datasets.
+
+PhyioNet 2017 The 2017 PhysioNet/CinC Challenge aims to classify, from 8,528 single-lead ECG recordings (between 30 s and 60 s in length), whether the recording shows normal sinus rhythm, atrial fibrillation (AF), an alternative rhythm, or is too noisy to be classified, i.e., four classes. We normalize the signals to have zero mean and unit standard deviation. Additionally, we zero-pad the shorter recordings to ensure they have the same length. We split the dataset into training, validation, and test sets according to the patients using a 60, 20, 20 configuration.
+
+For both datasets in CVD task, we used an 8-block ResNet model with a stride of 2, a learning rate of $5e - 4$ , and a batch size of 32. The learning rate for the guidance network was set same as the main classifier architecture.
+
+# C.1.4 STEP COUNTING
+
+The Clemson dataset has 30 participants (15 males, 15 females), Each participant wore three Shimmer3 sensors. We used the IMU sensor readings from non-dominant wrists to predict step count where each sensor recorded accelerometer and gyroscope data at $15\mathrm{Hz}$ . We calculated the total magnitude of the accelerometer and fed it to the model as a pre-processing without any filtering. We used window lengths of 32 seconds without an overlap in the regular walking setting. We conducted 10-fold cross-validation, with each fold consisting of 3 subjects for testing and validation. And, six randomly selected subjects were used for training in each fold. We used a 3 layer of FCN architecture, a learning rate of $5e - 4$ , and a batch size of 64 . The learning rate for the guidance network was set same as the main classifier architecture.
+
+# C.1.5 SLEEP STAGE CLASSIFICATION
+
+We used the Sleep-EDF dataset which has five classes: wake (W), three different non-rapid eye movements (N1, N2, N3), and rapid eye movement (REM). The dataset includes whole-night PSG sleep recordings, where we used a single EEG channel (i.e., Fpz-Cz) with a sampling rate of $100\mathrm{Hz}$ . We employed the identical data split as presented in the paper (Eldele et al., 2021), accessible online, without applying any additional pre-processing steps. We used a 16-block ResNet model with a stride of 2, a learning rate of $1e - 3$ and a batch size of 64. We ran three distinct seeds using the same split and reported the mean and standard deviation on the test set.
+
+# C.1.6 LUNG SOUND CLASSIFICATION
+
+We used Respiratory@TR which contains lung sounds recorded from left and right sides of posterior and anterior chest wall and back using two digital stethoscopes from 42 subjects collected in Antakya State Hospital (Altan et al., 2017). The 12 channels of lung sounds are focused on upper lung, middle lung, lower lung and costophrenic angle areas of posterior and anterior sides of the chest. The recordings are validated and labeled by two pulmonologists evaluating the collected chest X-ray, PFT and auscultation sounds of the subjects. Labels fall into 5 COPD severities (COPD0, COPD1, COPD2, COPD3, COPD4). The patients aged 38 to 68 are selected from different occupational groups, socio-economic status and genders. We performed 10-fold cross-validation on data from 42 subjects, with one fold reserved for validation in each iteration. All 12 channels, combining left and right chest wall recordings from six channels each, were used.
+
+The audio was segmented into 8-second windows with a 2-second overlap to capture temporal patterns. We used an 8-block ResNet model with a stride of 3, a learning rate of $1e - 5$ , and a batch size of 15. Although audio models typically use Mel-frequency spectrograms with different networks, we did not observe a significant performance difference between these models and our initial model. Therefore, we used the original model with temporal data, consistent with our main experiments.
+
+# C.2 METRICS
+
+We used the common evaluation metrics in the literature for each task. Specifically, we used mean absolute error (MAE), root mean square error (RMSE), and Pearson correlation coefficient $(\rho)$ for heart rate prediction. We used accuracy (Acc), macro-F1 score (F1) for activity recognition, and an additional area under the receiver operating characteristic curve (AUC) for cardiovascular disease classification (Kiyasseh et al., 2020). We used the mean absolute percentage error (MAPE) for step counting (Yang et al., 2020; Femiano et al., 2022). For lung audio classification, we evaluated performance using accuracy, macro F1, and weighted F1 scores (F1, W-F1). For sleep stage classification, we used the same metrics—accuracy, macro F1, and weighted F1 scores (F1, W-F1)—along with Cohen's Kappa coefficient $(\kappa)$ (Cohen, 1960).
+
+# C.3 BASELINES
+
+# C.3.1 LPF (BLURRING)
+
+In convolutional neural networks, pooling layers or convolutions with strides greater than 2 would conduct downsampling on the feature maps to reduce their size. However, since features are averaged or discarded in the downsampling, information may be lost, i.e., aliasing. Traditional pooling aggregates all values within the window to a single value. In contrast, Zhang (2019) aims to minimize information loss caused by pooling via replacing the traditional kernel with a Gaussian kernel as an low-pass filtering (LPF). Specifically, LPF applies a Gaussian-weighted function to the neighborhood values around each feature for convolution, and obtains a weighted average result.
+
+Compared to the pooling operations of simply selecting the maximum/average value within the window, LPF uses Gaussian-weighted averages to preserve the relative positions and spatial relationships between features. In our implementations, we used 1D version of LPF with a length of 5. We evaluated filter lengths of 3, 5, 7 to determine the optimum size for each task. During our experiments, we observed the best performance with a filter length of 5, except for HR prediction, where a length of 3 was optimal.
+
+# C.3.2 APS
+
+To make the sampling layer invariant to shifts, Chaman & Dokmanic (2021) proposed to subsample by partitioning feature maps into polyphase components and select the component with the highest norm. This approach has a significant limitation when applied to time series, especially with the nonlinear activation functions. It tends to overlook variations in boundaries arising from the translation of samples, thereby imposing additional shift constraints (Rojas-Gomez et al., 2022). Consequently, the evaluation of these methods is restricted to a narrow range of shifts, covering only a limited subset of the shift space. Moreover, as this approach selects the component with the highest norm, it requires feature maps to have unique values.
+
+# C.3.3 WAVELET NETWORKS
+
+Wavelet networks (Romero et al., 2024) consist of several stacked layers that respect scale and translation. At the beginning, the network consists of a lifting group convolution layer that lifts input time-series to the scale-translation group, followed by arbitrarily many group convolutional layers. At the end of the network, a global pooling layer is used to produce scale-translation invariant representations. Wavelet Networks are proposed for equivariant mappings between input and output. However, to turn an equivariant network into an invariant network, an extra layer that is equivariant in this degenerate sense (in practice, this often means either averaging or creating a histogram of the activations of the last layer) should be applied (Kondor & Trivedi, 2018). For example, the well-known wavelet scattering network achieves invariance by stacking equivariant layers followed by a final invariant one in that of scattering networks (Mallat, 2012). However, our proposed method does not require an additional layer as it operates on data manifolds directly.
+
+We used the original GitHub (https://github.com/dwromero/wavelet_networks) implementation, leaving the dropout and base parameters unchanged. We searched over the learning rate (e.g., $\{10^{-3}, 10^{-4}, 10^{-5}\}$ ) using the validation set for each time series task.
+
+# C.3.4 CANONICALIZATION
+
+We also compared our method with a canonicalization approach which is based on learning mappings to canonical samples. Specifically, we used the equiadapt library (https://github.com/arnab39/equiadapt) (Mondal et al., 2023; Kaba et al., 2023) while adding translation equivariant architectures for shift operations. For translation equivariant canonicalization architecture, we implemented a convolutional neural network with a kernel size of five and three layers. The network avoids pooling layers to prevent aliasing and instead employs global average pooling at the final stage to aggregate information across the signal length. The output is reshaped to separate the channels corresponding to each discrete translation, followed by aggregation over the fiber channels to produce translation-equivariant activations.
+
+We used a discrete Group representation with number of translations set to 16. The canonicalization network and classifier were trained jointly, incorporating a prior regularization loss to guide the learning process. For optimization, we employed the Adam optimizer with a learning rate of $1 \times 10^{-3}$ and no weight decay. We also performed a grid search to identify the optimal learning rate for the canonicalization network; however, no significant performance improvements were observed across different learning rates.
+
+# C.4 IMPLEMENTATION DETAILS
+
+Here, we have provided the details of the architectures, and hyperparameters. Primarily, we used the 1D ResNet (Hong et al., 2020) implementation in the supervised settings. While some alternative deep learning models can perform better in time series such as the combination of convolutional and LSTM layers, similar to previous works in shift consistency (Zhang, 2019), we focused on deep learning models which are mainly composed of convolutional layers.
+
+# C.4.1 ARCHITECTURES
+
+Here, we present the details of architectures that are investigated for the performance of shift-invariant techniques. Some details that are not given in the tables are as follows. Batch normalization (Ioffe & Szegedy, 2015) is applied after each convolutional block. ReLU activation is employed following batch normalization, in line with (He et al., 2015). We also applied a Dropout (Srivastava et al., 2014) with 0.5 after each activation and before the convolutions. Finally, a global average pooling is implemented before the linear layers.
+
+Tables 9, 10a, and 10b give an overall for the architectures with the number of parameters for each dataset. From these tables, it can be observed that the number of parameters for the guidance network is much less than the main classifier, where the ratio is close to $\approx 2 - 4\%$ .
+
+The parameter count of the guidance network could be further reduced by selectively inputting only the important frequencies or the frequency band for each time series task, which can be determined using prior knowledge, rather than the entire spectrum. Nevertheless, for the sake of consistency, we
+
+Table 9: ResNet architecture
+
+| Repetition | Layer | Kernel Size | Output Size | Stride |
| 1 | Input (C,T) | - | (C, T) | - |
| 1 | Conv | (5, 1) | (64, T/2) | 1 |
| R | Residual Block | | | |
| Conv | (5, 1) | (128, T/4) | S |
| Conv | (5, 1) | (128, T/4) | 1 |
| 1 | Linear | - | (n_classes,,) | - |
| # Parameters for dataset (C,T, R, S) |
| IEEE SPC (C=1, T=200, R=8, S=2) | ≈210k |
| DaLiA (C=1, T=200, R=8, S=2) | ≈210k |
| Chapman (C=4, T=1000, R=8, S=2) | ≈197k |
| PhysioNet (C=1, T=6000, R=8, S=2) | ≈197k |
| UCIHAR (C=9, T=65, R=8, S=2) | ≈200k |
| HHAR (C=6, T=51, R=8, S=2) | ≈200k |
| Respiratory (C=12, T=6000, R=8, S=3) | ≈200k |
| Sleep (C=1, T=3000, R=16, S=2) | ≈3.2M |
+
+Table 10: The model topologies of the classifier $f_{C}$ and guidance network $f_{G}$
+(a) FCN architecture
+
+| Model | Text-Only Pre-training | Image-Text Pre-training | Multitask Fine-Tuning |
| BLIP3(Xue et al., 2024) | Fully pre-trained(Phi3-mini) | Re-warmupCaption and interleaved text-image data; no pure text | Re-warmup |
| Flamingo(Alayrac et al., 2022) | Fully pre-trained(closed model) | Re-warmupCaption and interleaved text-image data; no pure text | (Scipped) |
| IDEFICS(Laurençon et al., 2024) | Fully pre-trained(Mistral-7B-v0.1) | Re-warmupInterleaved text-image data; no pure text | Re-warmup |
| MM1(McKinzie et al., 2024) | Fully pre-trained(closed model) | Re-warmupVarious image-text ratios (100:0, 91::9, 86::14, 66::33) | Re-warmup |
| DeepSeek-VL / DeepSeek-VL2(Lu et al., 2024; Wu et al., 2024b) | Fully pre-trained(DeepSeek) | Two-stage (re-warmup then re-warmup)Multitask; 30::70 image-text ratio | Re-warmup |
| Qwen-VL(Bai et al., 2023) | Fully pre-trained(Qwen) | Two-stage (re-warmup then re-warmup)Caption then multitask; no pure text | Re-warmup |
| PaliGemma(Beyer et al., 2024) | Fully pre-trained(Gemma) | Two-stage (re-warmup then continue LR schedule)Multitask; no pure text | Continued LR schedule |
| Janus / Janus Pro(Wu et al., 2024a; Chen et al., 2025) | Fully pre-trained(DeepSeek-LLM) | Two-stage (re-warmup then re-warmup)Multitask; no pure text, followed by 30::70 image-text ratio | No warmup |
| Prismatic(Karamcheti et al., 2024) | Fully pre-trained(LLaMA) | No image-text pre-training. | Re-warmup |
| Ours | Partially pre-trained(OpenLM) | Continued LR scheduleVarious image-text ratios | Re-warmup |
+
+to a model which had already been pre-trained. In practice, this is implemented by resuming from the same learning rate schedule (Section 2.1.2). A more detailed comparison overview of different models can be found in Table 1.
+
+# 2.1.1 TEXT-ONLY PRE-TRAINING
+
+During the initial pre-training process, only the LLM backbone is trained with the training recipe from Gadre et al. (2024b). Rather than re-training from scratch, we instead leverage existing open checkpoints from DCLM-1B (Li et al., 2024b) $^{1}$ . This is a 1.4B parameter model trained on text tokens from DCLM-Baseline mixed with StarCoder (Li et al., 2023b) and ProofPile (Azerbayev et al., 2024), for a total of 4.3T tokens, and it outperforms the recent LLaMA-3.2 1B model $^{2}$ on several commonly used text benchmarks.
+
+The pre-training learning rate schedule is warmup-cosine with a peak learning rate of $10^{-2}$ and a final learning rate of $10^{-5}$ . In addition to using the final model, we also use checkpoints from $20\%$ , $40\%$ , $60\%$ , and $80\%$ through training, which correspond to 860B, 1.72T, 2.58T, and 3.44T training tokens respectively. Since these checkpoints are taken from the middle of training, their learning rates are higher than the typical last learning rates of most released model weights, allowing us to effectively resume pre-training (Figure 3).
+
+Additional hyperparameters for the text-only pre-training stage can be found in Appendix A.
+
+# 2.1.2 IMAGE-TEXT PRE-TRAINING
+
+We take checkpoints from the text-only pre-trained model and resume the cosine learning rate schedule with a cooldown phase, as illustrated in Figure 3.
+
+For our experiments, we perform multi-modal training using both caption data and task-related instruction data. As opposed to other VLMs like MM1 (McKinzie et al., 2024), BLIP3 (Xue et al., 2024), and IDEFICS (Laurençon et al., 2024) which use interleaved data in addition to caption data, we limit our setting to either text only or text + a single image. Our minimal setting allows us to control as many factors as possible to isolate the most fundamental factors to train a performant multi-modal model.
+
+
+Figure 3: Representation of the different learning rate schedules used for our experiments. 'Main schedule' corresponds to the learning rate for the initial, text-only pretraining. Other colored schedules are the ones used for image-text training and extend over 28B tokens each. They have been upscaled and appear as extending over 280B tokens for readability.
+
+There are a number of design choices involved in this stage. We give an overview of these design choices below:
+
+Text dataset selection: During the image-text pre-training stage, we train with the DCLM-baseline dataset, mixed with StarCoder and MathPile. This is selected to match the dataset used during the text-only pre-training step to ensure continuity from the LLM checkpoints that we resume.
+
+Caption dataset selection: We train with the DataCompDR-1B caption dataset (Vasu et al., 2024), which is an enhancement over DataComp-1B (Gadre et al., 2024a) by regenerating higher quality captions. To arrive at this selection, we ablated over several caption datasets (Appendix J.1).
+
+Image-text ratio: The effect of mixing text and images has not been thoroughly documented and is one of the key questions we explore in this paper. McKinzie et al. (2024) show that adding some text tokens helps with few-shot and text-only performance, though they show only up to $33\%$ text and operate at slightly different settings (fully pre-trained vs partially pre-trained model). DeepSeek-VL (Lu et al., 2024) uses a fixed $70\%$ text ratio but does not discuss other ratios. We conduct a thorough investigation on the image-text ratio in Sections 3.3 and 3.2.
+
+Amount of training: We train all our models with a token multiplier of 20, following approximate Chinchilla optimal scaling (Hoffmann et al., 2022). For a 1.4B model, this equates to around 28B tokens. From our experiments at smaller scale, we observe that trends generally hold across higher token multipliers (Appendix G).
+
+Image encoder selection: We found the SigLIP encoder to work well for our setting, which confirms findings from Beyer et al. (2024). The DINO+SigLIP combination from Karamcheti et al. (2024) gave slightly better results on some benchmarks; for simplicity, however, we use SigLIP for most of our experiments. See Appendix J.2 for full ablations.
+
+Frozen encoder weights: The vision encoder can be kept frozen (Karamcheti et al., 2024) (i.e., its parameters are not trained) or trained end-to-end with the language model (Tong et al., 2024; Beyer et al., 2024). In our small-scale ablations, we found it more effective to freeze the image encoder weights in both the pre-training and fine-tuning stages (Appendix I), so we freeze the weights for the rest of this section.
+
+Each image is encoded to 729 tokens, and we train all models for a sequence length of 1024. We provide additional implementation details in Appendix C.
+
+# 2.1.3 INSTRUCTION FINE-TUNING
+
+We fine-tune our pre-trained models with the LLaVA dataset (Liu et al., 2024) using the Prismatic framework (Karamcheti et al., 2024). For each model trained in the previous step, we fine-tune for $\{1,2,3,4\}$ epochs. We use a cosine learning rate schedule with warmup. To account for different epoch numbers having different learning rate schedules, we treated each epoch number as its own separate run. A full list of hyperparameters can be found in Appendix D.
+
+# 2.2 EVALUATION
+
+We want our VLMs to perform well in vision-language tasks while still retaining their performance in text-only tasks. As such, we conduct evaluations over a suite of diverse tasks, both image-based and text-based. In our early experiments, we trained a 79M parameter model with text and used it to search the training space before scaling to the 1B experiment presented in this work. We only keep the downstream text tasks where a 79M parameter model performs better than random chance. We test our model without any in-context learning, thus all results reported are 0-shot.
+
+Vision-language tasks: We evaluate on a subset of vision-language tasks used by Karamcheti et al. (2024) $^3$ . These tasks test general visual reasoning, spatial reasoning, ability to read text, object localisation and hallucination. For a full set of vision-language evaluation tasks, see Appendix E.
+
+In addition to reporting the scores on individual tasks, we also report an aggregate metric. All our results are accuracy scores (between [0, 1]). To aggregate them, we subtract the random baseline accuracy, then average the result over all tasks. We call this aggregate metric the stable score. Its absolute value can be interpreted as the percentage points of advantage over random guessing. Its trend can be understood as average percentage points on the suite of benchmarks.
+
+Text tasks: We evaluate on a suite of tasks taken from Gadre et al. (2024b) and conduct our evaluations with Eleuther's LM Harness (Gao et al., 2024). These tasks test for general and specific knowledge, reasoning, and commonsense. We also compute the stable score as previously described. For a full list of our text evaluation tasks, see Appendix E.
+
+# 3 RESULTS
+
+This section presents a series of experiments exploring different aspects of multi-modal training. We aim to find important choices that affect the downstream performance of the model on a set of diverse visual tasks and text-only question answering.
+
+All experiments below are conducted on the 1.4 billion parameter regime. We make use of various checkpoints of the DCLM-1B model along its text-only pre-training. This is explored in more detail in Section 3.1 below.
+
+All experiments use in different proportions three dataset sources. (1) A text-only dataset, DCLM-baseline (Li et al., 2024b), which was used to pre-train the 1B language model. (2) An image caption dataset, DataCompDR-1B (Vasu et al., 2024), and (3) An image instruction tuning dataset, mix of openly available data described in A.1/Multimodal Instruct Tuning (Karamcheti et al., 2024).
+
+Each section below discusses a different set of experiments with a given setup summarized in a green box labeled Setup. The setup describes which checkpoint of the text-only pre-trained model is used, the mix of text and image captioning data used for image-text pre-training, and the number of fine-tuning epochs performed on the image instruction tuning dataset. Within each description, we use $x$ to denote the variable that we are changing for that particular experiment, while keeping the rest of the variables constant.
+
+For each of the sections below, there will be two plots, each with two vertical axes, representing our experiment results on vision-language and text tasks. Each point on a plot represents a 1B model initialized from one of the text-only pre-trained checkpoints, pre-trained with text and image captions, and fine-tuned with image instruction tuning data. The plot on the left represents the stable score, an aggregate of performance results, across the vision benchmark on one hand and text benchmarks in the other. The plot on the right represents one of each set of benchmark, the VQA-v2 (Goyal et al., 2017) results and the ARC-easy (Clark et al., 2018) results. These might be easier to interpret but do not convey the full extent of the evaluation. The full list of benchmarks as well as detailed graphs for the vision performance can be found in Appendix F.
+
+# 3.1 THE IMPACT OF AMOUNT OF TEXT-ONLY PRE-TRAINING
+
+# Setup
+
+- $x\%$ checkpoints of 1B model
+- Image-text pre-trained for 28B tokens:
+90% Text-only
+$\circ 10\%$ Image captions
+- Fine-tuning 4 epochs
+
+For this experiment, we use different checkpoints along the text-only pre-training. The initial text model was trained with a warmup-cosine learning rate schedule. All continued image-text pre-training runs follow the learning rate schedules represented in Figure 3. We continue training starting from the last text-only learning rate value, if not too low, and continue training for 28B. For $0\%$ and $100\%$ the learning
+
+rate would be too low, thus we adopt a linear warmup-cosine decay with a maximum learning rate of $3 \times 10^{-3}$ . See Figure 3.
+
+Figure 4 below shows the progression of the model performance as a function of the text pre-training completion of the initial checkpoint. We see that up to $80\%$ , a larger amount of text-only pre-training improves performance for both text and vision tasks, suggesting that it's helpful to start with a stronger model which has been trained for a greater number of text tokens. However, at $100\%$ the learning rate schedule is different and could affect the results. Many similar works all initialize their model to a $100\%$ text-only pre-trained model and all re-warmup the learning rate during the image-text pre-training phase. From our experiments, we observe a decrease in performance at $100\%$ , suggesting that continued training is preferable to re-training a $100\%$ fully pre-trained text model.
+
+
+Figure 4: Varying the length of text-only pre-training. We analyze the impact of adding image data after varying amounts of text-only pre-training, showing results on vision benchmarks (green) and text benchmarks (blue). On the left, we show results across a suite of vision and text benchmarks; on the right, we plot two common benchmarks, VQA-v2 and ARC-easy. Introducing images at around $80\%$ of the way through training performs best, maintaining high vision and text task performance. Note: The points at $100\%$ are marked with hollow circles to highlight that they are trained with a different learning rate schedule, as shown in Figure 3
+
+
+
+Longer text pre-training improves performance but, when adding image input, continued training gives better performance than re-warming up.
+
+# 3.2 THE IMPACT OF ADDING IMAGES BEFORE THE END OF PRE-TRAINING
+
+# Setup
+
+- $80\%$ pre-trained 1B model
+- Image-text pre-trained for 28B tokens:
+$1 - x\%$ text
+$0x\%$ image captions
+- Fine-tuning 4 epochs
+
+Building on the results from Figure 4, we take the $80\%$ checkpoint and vary the image-text ratio we add during pre-training. Figure 5 shows the evolution of the model performance as we pre-train (from a text-only pre-trained model) with more image data. We keep the total number of pre-training tokens constant at 28B, adjusting only the image-text ratio.
+
+From Figure 5, we see that as more image data is
+
+added to the mix, better downstream performance is obtained on vision tasks. However, text-only downstream performance decreases. When only image data is used for image pre-training, both text and vision task performance drop significantly. We observe that from $0\%$ images (i.e. train on $100\%$ text) to $1\%$ images, there is a slight drop in vision stable score performance. We believe this can possibly be attributed to the model being confused at the addition of this new task, and that it requires
+
+some minimum number of tokens in order to truly learn the image representations. Interestingly, performance on text-only tasks gets slightly better when some image data is added to the pre-training mix.
+
+Introducing image data at a late stage during pre-training improves vision performance while maintaining text performance.
+
+
+Figure 5: Varying the ratio of image to text data, after some text-only pretraining. We analyze the impact of the ratio of image to text data in pre-training, after the model has seen text-only data for most of pre-training (80%). Unlike when training from scratch (Figure 6), we find that adding vision data significantly helps vision performance, while maintaining high text accuracy.
+
+
+
+# 3.3 THE IMPACT OF ADDING IMAGES WHEN TRAINING FROM SCRATCH
+
+# Setup
+
+- $0\%$ randomly initialized 1B model
+- Image-text pre-trained for 28B tokens:
+
+$1 - x\%$ text
+$0x\%$ image captions
+- Fine-tuning 4 epochs
+
+is section mirrors Section 3.2, with the only range being starting the image-text pre-training from scratch as opposed to from an $80\%$ checkpoint. Figure 6 shows the evolution of the model performance as we pre-train the model from scratch with pre-image data.
+
+We observe slightly different trends for models
+
+trained from scratch (Figure 6) as compared to the models trained from the $80\%$ checkpoint (Figure 5). Here, as more image data are added to the mix, we obtain worse downstream performance on both text and vision tasks. These models were trained for 28B tokens each and would likely benefit from more training; it is possible that at larger scales, these trends will more closely resemble those in Figure 5. Interestingly, just like in Figure 5, adding $1\%$ image data strongly degrades the ability of the model to be fine-tuned for vision tasks.
+
+
+Figure 6: Varying the ratio of image to text data, when training from scratch. We analyze the impact of the image-text ratio in pre-training from scratch without any language-only pre-training. Perhaps surprisingly, when training from scratch, adding vision data consistently hurts both vision and text performance, suggesting a period of language-only training early on is important for VLMs.
+
+
+
+Introducing image data when training from scratch at small scales does not improve downstream vision performance.
+
+# 3.4 THE IMPACT OF INSTRUCTION TUNING DATA IN PRE-TRAINING
+
+# Setup
+
+- $80\%$ pre-trained 1B model
+- Image-text pre-trained for 28B tokens:
+
+90% text
+$0 - x\%$ image captions
+$0x\%$ image instruction tuning data
+- Fine-tuning for 4 epochs
+
+We train a 1B model with a mix of $10\%$ image-text and $90\%$ text-only data and vary the source of the image-text data from purely image captions to purely instruction tuning data. Figure 7 shows the evolution of the model performance as we increase the proportion of image instruction tuning in the mix of image-text data. We observe that the best performance on vision-language tasks occurs when there are no instruction tuning data included in the pre
+
+training mix, although adding instruction-tuning data seems to positively affect scores on text-only tasks. From a vision-language standpoint, we conclude that image instruction tuning data should not be added to the pre-training mix and only be used in a separate fine-tuning stage.
+
+
+Figure 7: Varying the proportion of instruction tuning data in the image mix. Is including instruction tuning data during pre-training is helpful for VLMs? Surprisingly, we find that adding this data to pre-training harms performance. We hypothesize that this may be due to overfitting, or because mixing instruction tuning data with image-caption pairs degrades learning at this scale.
+
+
+
+The presence of image instruction tuning data in pre-training harms downstream evaluations.
+
+# 3.5 THE IMPACT OF FINE-TUNING ON VISION AND TEXT PERFORMANCE
+
+# Setup
+
+- $80\%$ pre-trained 1B model
+- Image-text pre-trained for 28B tokens: $0.90\%$ text
+
+$\circ 10\%$ image captions
+- Fine-tuning for $x$ epochs
+
+Figure 8 shows the evolution of the model 1B performance as it is fine-tuned for 1 to 6 epochs. Looking at the stable score, we observe that up to 4 epochs, downstream vision task performance improves and the downstream text performance degrades. Beyond 4 epochs of fine-tuning, both vision and text performance degrade, likely due to overfitting.
+
+
+Figure 8: Varying the number of fine-tuning epochs. We find that fine-tuning for 2-4 epochs after pre-training performs best for vision tasks, with 2 epochs being a sweet spot for maintaining text performance while achieving high vision performance.
+
+
+
+Fine-tuning for image instruction improves vision-language tasks even beyond one epoch, up to a point, but degrades text-only task performance.
+
+# 4 RELATED WORK
+
+The related work covers three subjects, first, two-stage vision language models (VLMs), which rely on pre-training with text-only data, then recent VLMs which introduce vision data earlier in pre-training, and finally, multimodal datasets for VLM training.
+
+Two-stage VLMs. The standard approach for training vision language models involves first training a large language model on text-only data, and then incorporating vision data in a second fine-tuning phase. This strategy is employed effectively by models including Liu et al. (2024; 2023) (LLaVA and variants), Alayrac et al. (2022) (Flamingo), Li et al. (2023a); Xue et al. (2024) (BLIP-2 and BLIP-3), Laurençon et al. (2024), Laurençon et al. (2023) (IDEFICS and Obelics), Chen et al. (2025) (Janus and Janus-Pro) among others. McKinzie et al. (2024) explore design choices for training VLMs complementary to the ones we assess, namely the vision-language connector choice, the pretraining strategy for the image encoder, and the image resolution, while using a two-stage training strategy with a fixed, pre-trained language model.
+
+Pre-training with visual data. Bavishi et al. (2023) (Fuyu-8B) is a notable exception which trains a vision-language model from scratch, but the training strategy is not publicly released. Team (2024) introduces a series of early-fusion VLMs, which, similar to our work, introduce image data during pre-training, but choose a specific, handcrafted strategy for training. For multi-modal pre-training, Aghajanyan et al. (2023) find scaling laws for pre-training from text, code, image-text, image and speech data in terms of unimodal and bi-modal combinations. They investigate training dynamics for various mixtures of pre-training data and model sizes in terms of perplexity. Their study is related to ours in the sense of pre-training with mixtures of data, but it differs in that it does not involve any downstream evaluations. We focus specifically on the setting of VLM training, and thoroughly investigate the results on standard benchmarks.
+
+Datasets Collecting image-text pre-training and instruction data is challenging, due to the low quality of image-text pairs on the internet. Most datasets use a combination of curation and re-categoration to address this, including Desai et al. (2021) (RedCaps), or Thomee et al. (2015) (YFCC-100M). In this work, we specifically rely on recent large-scale, high-quality datasets that perform well for training multimodal models: Changpinyo et al. (2021) (ConceptualCaptions-12M) follows the ConceptualCaptions-3M (Sharma et al., 2018) pipeline, curating images and alt-text pairs from the internet, with relaxed filters to increase the dataset size. Gadre et al. (2024a) (DataComp-1B) uses CLIP-based filtering to curate a high-quality set of 1 billion image-text pairs, and Vasu et al. (2024) (DataCompDR-1B) uses a vision-language model to generate synthetic captions for the original DataComp-1B dataset. Li et al. (2024c) (Recap-DataComp-1B) further improves on this by using a LLaMA3-based LLaVA-1.5 model (Liu et al., 2023) to perform the recategoriation. For finetuning, we use the LLaVA-1.5 (Liu et al., 2024; 2023) mix, which consists of high quality, curated image datasets, along with instruction-tuning data generated using a large language model. Recent datasets further explore interleaved data, where the same sequence counts several images and text piece that relate to each other. Interleaved datasets include OBELICS (Laurençon et al., 2023), the Cauldron (Laurençon et al., 2024), and MINT-1T (Awadalla et al., 2024). We leave the analysis of this data for VLM pretraining to future work.
+
+# 5 CONCLUSION
+
+In this paper, we address some key gaps and challenge several common assumptions in vision-language model (VLM) pre-training. Specifically, our work questions the traditional practice of separating text and image pre-training phases, demonstrating that a more integrated approach of incorporating image data during pre-training can yield superior downstream results. We recommend future VLM efforts leverage intermediate pre-training checkpoints to incorporate images before completing the pre-training process. In this setting, it's important to carefully manage the image-to-text ratio: unlike the typical approach where a separate image training stage is done with $100\%$ caption images, this integrated setting requires a balanced image-text ratio to avoid performance degradation. We plan to make our code and our testbed of models publicly available, and we hope that our findings will provide a strong empirical foundation for open-source VLM pre-training.
+
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+
+# A TEXT-ONLY PRE-TRAINING HYPERPARAMETERS
+
+We preformed initial exploratory experiments with a 79M parameter model and scale our final experiments to a 1B model with settings from Gadre et al. (2024b) given in Table 2. Our text models are trained with the OpenLM (Gururangan et al., 2023) codebase for next token prediction on the DCLM-baseline dataset (Li et al., 2024b).
+
+Table 2: The two models and set of hyperparameters used in our experiments. Models have number of parameters $N$ , with number of layers $n_{\text{layers}}$ , number of attention heads $n_{\text{heads}}$ , model width $d_{\text{model}}$ , and width per attention head $d_{\text{head}}$ . Batch sizes are global and in units of sequences. Each sequence has 2,048 tokens. A100 GPU hours are at $M = 150$ . For the 1.4B scale, a batch size of 256 performs slightly better than 512.
+
+| Type | Model | # Params | POPE↑ | MME↑ | Flickr30k↑ | VQAv2(test)↑ | GQA↑ | MMMU↑ |
| Und. Only | LLaVA-v1.5 (Liu et al., 2024b) | 7B | 85.9 | 1510.7 | - | 78.5 | 62.0 | 35.4 |
| InstructBLIP (Dai et al., 2023) | 13B | 78.9 | 1212.8 | - | - | 49.5 | - |
| Qwen-VL-Chat Bai et al. (2023) | 7B | - | 1487.5 | - | 78.2 | 57.5 | - |
| mPLUG-Owl2 (Ye et al., 2024b) | 7B | 85.8 | 1450.2 | - | 79.4 | 56.1 | - |
| LLaVA-v1.5-Phi-1.5 | 1.3B | 84.1 | 1128.0 | 69.6 | 75.3 | 56.5 | 30.7 |
| Und. and Gen. | Gemini-Nano-1 (Anil et al., 2023) | 1.8B | - | - | - | 62.7 | - | 26.3 |
| CoDI (Tang et al., 2024) | - | - | - | 12.8 | - | - | - |
| Emu (Sun et al., 2023c) | 13B | - | - | 77.4 | 57.2 | - | - |
| NExT-GPT (Wu et al., 2023b) | 13B | - | - | 84.5 | 66.7 | - | - |
| SEED-X (Ge et al., 2024) | 17B | 84.2 | 1435.7 | 52.3 | - | 47.9 | 35.6 |
| DreamLLM (Dong et al., 2024) | 7B | - | - | - | 72.9 | - | - |
| VILA-U (Wu et al., 2024) | 7B | 85.8 | 1401.8 | - | 79.4 | 60.8 | 31.6 |
| Emu3 (Wang et al., 2024) | 8B | 85.2 | - | - | 75.1 | 60.3 | - |
| Chameleon (Team, 2024) | 34B | - | - | 74.7 | 66.0 | - | - |
| Show-o (Ours) | 1.3B | 80.0 | 1097.2 | 62.5 | 69.4 | 58.0 | 26.7 |
| Show-o‡ (Ours) | 1.3B | 84.5 | 1232.9 | 67.6 | 74.7 | 61.0 | 27.4 |
+
+$\mathbf{v} = \{v_{1}, v_{2}, \dots, v_{N}\}$ for multimodal understanding, we maximize the likelihood of text tokens by employing the standard language modeling objective:
+
+$$
+\mathcal {L} _ {\mathrm {N T P}} = \sum_ {i} \log p _ {\theta} \left(v _ {i} \mid v _ {1}, \dots , v _ {i - 1}, u _ {1}, \dots , u _ {M}\right), \tag {1}
+$$
+
+where $p(\cdot |\cdot)$ indicates the conditional probability which is modeled by the weights $\theta$ of Show-o and stochastic gradient descent is used to train the model. Note that, if the input sequence involves only text tokens, there are no conditional terms on image tokens $\mathbf{u} = \{u_{1}, u_{2}, \dots, u_{M}\}$ .
+
+With the proof in Appendix A, we seamlessly integrate the simplified discrete diffusion modeling within Show-o by employing the mask token prediction as a learning objective. Hence, for modeling image tokens $\mathbf{u} = \{u_1, u_2, \dots, u_M\}$ within the input sequence, we first randomly replace the image tokens with the [MASK] token, notated as $u_*$ , at a random ratio (controlling by a time step) to create a masked sequence $\mathbf{u}_* = \{u_*, u_2, \dots, u_*\}$ . An illustration can be found in Fig. 9. Next, we aim to reconstruct the original image token from the masked tokens conditioning on unmasked regions and preceding text tokens by maximizing the following likelihood:
+
+$$
+\mathcal {L} _ {\mathrm {M T P}} = \sum_ {j} \log p _ {\theta} \left(u _ {j} \mid u _ {*}, u _ {2}, \dots , u _ {*}, u _ {M}, v _ {1}, \dots , v _ {N}\right). \tag {2}
+$$
+
+Note that the loss is only applied to the masked tokens. Specifically, we follow the sampling strategy used by MaskGIT Chang et al. (2022; 2023) to mask image tokens and reconstruct them via the information from all text and unmasked image tokens within the input sequence. Following the classifier-free guidance introduced by Ho & Salimans (2022), we randomly replace the conditioned text tokens using a null text “” with some probability.
+
+Given a batch size of input sequences, the overall training loss is the combination of $\mathcal{L}_{\mathrm{MTP}}$ and $\mathcal{L}_{\mathrm{NTP}}$ :
+
+$$
+\mathcal {L} = \mathcal {L} _ {\mathrm {M T P}} + \alpha \mathcal {L} _ {\mathrm {N T P}}, \tag {3}
+$$
+
+where $\alpha$ is the hyper-parameter weighting the loss term $\mathcal{L}_{\mathrm{NTP}}$ . The training schedule mainly involves three stages, and we provide more details in Appendix C.
+
+Inference Stage. In multimodal understanding, given an image accompanying visual questions, Show-o autoregressively predicts textual answers. In visual generation, we use all [MASK] tokens as initial input for Show-o, in which [MASK] tokens will be iteratively replaced by the predicted image tokens within $T$ steps. More inference details are provided in Appendix D.
+
+# 4 EXPERIMENTS
+
+# 4.1 EXPERIMENTAL SETUP
+
+Datasets. We assemble two scales of datasets, i.e., around 35M and 2.0B image-text pairs, and collect around 2M high-quality data for multimodal understanding and generation fine-tuning. Besides,
+
+
+User: Do you think the image is unusual or not?
+Figure 6: Comparisons of VQA capabilities between Chameleon and Show-o.
+
+# Show-o:
+
+The image is unusual because it depicts a living room with a couch and a table in the middle of the ocean, surrounded by a large body of water. This is not a typical setting for a living room, as living rooms are usually indoors and designed for relaxation and entertainment. The presence of the couch and table in the ocean adds a unique and unexpected element to the scene, making it stand out from typical living room setups.
+
+# Chameleon:
+
+The image is unusual because of the transparent and colorful fish and coral in the water. The image is also unusual because of the clear blue sky with white clouds. The unusual objects in the image are the fish and coral in the water. The unusual objects in the image are the fish and coral in the water. The unusual objects in the image are the fish and coral in the water. The unusual objects in the image are the fish and coral in the water.
+
+RefinedWeb (Penedo et al., 2023) is adopted as text corpora to maintain the language modeling capability. Appendix E provides more details about these datasets.
+
+Evaluation Details. Following LLaVA (Liu et al., 2024b), we evaluate the multimodal understanding capabilities of Show-o on POPE, MME, Flickr30k, VQAv2, GQA, and MMMU benchmarks. Besides, we adopt Fréchet Inception Distance (FID) on MSCOCO dataset to evaluate the generation fidelity of Show-o. Further, we follow SD3 (Esser et al., 2024) to evaluate the text-to-image generation capabilities of Show-o on the GenEval (Ghosh et al., 2023) benchmark.
+
+Implementation details. Current version of Show-o is based on Phi-1.5 (1.3B) (Li et al., 2023). In the following, the default Show-o employs discrete image tokens as input for both multimodal understanding and generation. Show- $\mathbf{o}^{\dagger}$ and Show- $\mathbf{o}^{\ddagger}$ indicate the use of continuous image representations from the pre-trained MAGVIT-v2 and CLIP-ViT (corresponding to options (b) and (c) in Fig. 3), respectively, for multimodal understanding. Training details can be found in Appendix F.
+
+# 4.2 MULTIMODAL UNDERSTANDING
+
+Quantitative Evaluation. Table 1 presents the multimodal understanding capability of Show-o on public benchmarks, such as image captioning and visual question-answering tasks. i) The current version of Show-o is built upon Phi-1.5 and thus we follow LLaVA to train Show-o's understanding only counterpart as our direct baseline, namely LLaVA-v1.5-Phi-1.5. The proposed Show-o exhibits comparable performance in all evaluation metrics to the baseline LLaVA-v1.5-Phi-1.5, which is dedicated and optimized to only multimodal understanding. This demonstrates the great potential of our framework to unify multimodal understanding and generation in one single transformer. ii) When comparing with understanding only models including InstructBLIP, Qwen-VL-Chat, and mPLUG-Owl2 on multimodal understanding, our model with a much smaller model size also achieves competitive performance on POPE, MME, Flickr30k and VQAv2 benchmarks and performs better on GQA benchmark. iii) Compared with unified models with a much larger number of parameters, such as NExT-GPT-13B and Chameleon-34B, our model also achieves decent performance on Flickr30k benchmark and performs much better on VQAv2 benchmark.
+
+Qualitative Results. We present Show-o's visual question-answering capability and make comparisons with Chameleon in Fig. 6. It is evident that when presented with a query image, Show-o can respond to commonly asked questions, even addressing the unusual aspects within the image. In the example of Fig. 6, when asked, "Do you think the image is unusual or not", Chameleon fails to correctly identify the unusual aspect. In contrast, Show-o's response, "as living rooms are usually indoors and designed for relaxation and entertainment", is more accurate.
+
+# 4.3 VISUAL GENERATION
+
+Results on MSCOCO 30K. We present zero-shot FID of Show-o on MSCOCO 30K in Table 2. It can be observed that, compared to generation models trained with larger numbers of parameters and training images such as GLIDE and DALL-E 2, Show-o achieves a better FID, i.e., 9.24, with only 1.3B parameters and 35M training data. Though GigaGAN, Imagen, and RAPHAEL obtain a relatively better performance
+
+Table 2: MSCOCO zero-shot FID. Und. and Gen. denote "understanding" and "generation", respectively.
+
+| Type | Method | # Params | # Images | FID-30K↓ |
| Gen. Only | DALL-E (Ramesh et al., 2021) | 12B | 250M | 27.50 |
| GLIDE (Nichol et al., 2021) | 5B | 250M | 12.24 |
| LDM (Rombach et al., 2022) | 1.4B | 400M | 12.64 |
| DALL-E 2 (Ramesh et al., 2022a) | 6.5B | 650M | 10.39 |
| SDv1.5 (Rombach et al., 2022) | 0.9B | 2000M | 9.62 |
| GigaGAN (Kang et al., 2023) | 0.9B | 2700M | 9.09 |
| PixArt (Chen et al., 2024) | 0.6B | 25M | 7.32 |
| Imagen (Saharia et al., 2022) | 3B | 860M | 7.27 |
| RAPHAEL (Xue et al., 2024) | 3B | 5000M+ | 6.61 |
| Und. and Gen. | CoDI (Tang et al., 2024) | - | 400M | 22.26 |
| LWM (Liu et al., 2024a) | 7B | - | 12.68 |
| SEED-X (Ge et al., 2024) | 17B | - | 14.99 |
| DreamLLM (Dong et al., 2024) | 7B | - | 8.76 |
| Show-o (Ours) | 1.3B | 35M | 9.24 |
+
+Table 3: Evaluation on the GenEval (Ghosh et al., 2023) benchmark. Und. and Gen. denote "understanding" and "generation", respectively. We highlight the model size of Show-o in green, and we use blue to highlight the larger model size than ours. Obj.: Object. Attri.: Attribute.
+
+| Type | Method | # Params | Single Obj. | Two Obj. | Counting | Colors | Position | Color Attr. | Overall↑ |
| Gen. Only | LlamaGen (Sun et al., 2024) | 0.8B | 0.71 | 0.34 | 0.21 | 0.58 | 0.07 | 0.04 | 0.32 |
| LDM (Rombach et al., 2022) | 1.4B | 0.92 | 0.29 | 0.23 | 0.70 | 0.02 | 0.05 | 0.37 |
| SDv1.5 (Rombach et al., 2022) | 0.9B | 0.97 | 0.38 | 0.35 | 0.76 | 0.04 | 0.06 | 0.43 |
| PixArt-alpha (Chen et al., 2024) | 0.6B | 0.98 | 0.50 | 0.44 | 0.80 | 0.08 | 0.07 | 0.48 |
| SDv2.1 (Rombach et al., 2022) | 0.9B | 0.98 | 0.51 | 0.44 | 0.85 | 0.07 | 0.17 | 0.50 |
| DALL-E 2 (Ramesh et al., 2022a) | 6.5B | 0.94 | 0.66 | 0.49 | 0.77 | 0.10 | 0.19 | 0.52 |
| SDXL (Podell et al., 2023) | 2.6B | 0.98 | 0.74 | 0.39 | 0.85 | 0.15 | 0.23 | 0.55 |
| SD3 (d=24) (Esser et al., 2024) | 2B | 0.98 | 0.74 | 0.63 | 0.67 | 0.34 | 0.36 | 0.62 |
| Und. and Gen. | CoDI (Tang et al., 2024) | - | 0.89 | 0.16 | 0.16 | 0.65 | 0.02 | 0.01 | 0.31 |
| LWM (Liu et al., 2024a) | 7B | 0.93 | 0.41 | 0.46 | 0.79 | 0.09 | 0.15 | 0.47 |
| SEED-X (Ge et al., 2024) | 17B | 0.97 | 0.58 | 0.26 | 0.80 | 0.19 | 0.14 | 0.49 |
| Emu3 Wang et al. (2024) | 8B | - | - | - | - | - | - | 0.66 |
| Transfusion Zhou et al. (2024) | 7.3B | - | - | - | - | - | - | 0.63 |
| Chameleon (Team, 2024) | 7B | - | - | - | - | - | - | 0.39 |
| Show-o (Ours) | 1.3B | 0.98 | 0.80 | 0.66 | 0.84 | 0.31 | 0.50 | 0.68 |
| Show-o‡ (Ours) | 1.3B | 0.98 | 0.85 | 0.67 | 0.81 | 0.28 | 0.55 | 0.69 |
+
+than Show-o, they are much larger in model size (3B v.s. 1.3B) and trained with much more data. In comparison to unified models, Show-o also exhibits improvement. However, it is worth noting that FID on MSCOCO 30K may not be a comprehensively accurate assessment of generation fidelity. The reason lies in the fact that existing generation models are commonly fine-tuned with high-quality and aesthetic images that do not align with the distribution of the MSCOCO dataset.
+
+Results on GenEval. One can observe in Table 3 that when comparing to the model in a similar size such as LDM (1.4B), Show-o obtains significantly better performance in all six metrics, with an improvement of around 0.24 overall. Besides, Show-o achieves a better performance than DALL-E 2, which is 5 times larger in model size, and SDXL. Further, Show-o, with only 1.3B parameters, achieves comparable performance to models with around two times larger number of parameters such as SD3 (2B). It indicates that our unified model's generative capabilities are comparable to or even surpass those of specialized generation models. In comparison to unified models such as CoDI, SEED-X, and Chameleon, Show-o also demonstrates significant improvements.
+
+Qualitative Results. We show image samples generated by Show-o in Fig. 7. One can observe that Show-o is capable of generating diverse, interesting, and realistic visual content in a resolution of $512 \times 512$ . For example, Show-o can generate a futuristic style of car, a highly detailed face, cute objects, and vivid scenery with vibrant contrast.
+
+Text-guided Inpainting and Extrapolation. As mentioned, Show-o naturally supports text-based inpainting and extrapolation without requiring any fine-tuning. We illustrate examples in Fig. 8 (a). As shown on the top of the figure, given an input image and inpainting mask, Show-o can inpaint the original red trolley car to a blue sports car with sleek curves and tinted windows based on the user-provided text prompt. Specifically, we first tokenize the original image, mask those tokens to be inpainted, and then Show-o will gradually replace the masked tokens with predicted image tokens. Besides, Show-o is capable of extrapolating the original image horizontally/vertically based on the given text prompt. These cases significantly demonstrate the inherent advantages of Show-o over those autoregressive models for downstream applications.
+
+# 4.4 MIXED-MODALITY GENERATION OF VIDEO KEYFRAMES AND CAPTIONS
+
+Here, we explore the mixed-modality generation ability of Show-o based on the text descriptions and video keyframes in the GenHowTo dataset. Given a sequence of interleaved text descriptions and video keyframes (as shown at the bottom of Fig. 4), Show-o is trained to predict the next text tokens or keyframe tokens conditioning on all preceding tokens. Thus, Show-o can generate mixed-modality of text descriptions and video keyframes. Examining a single frame, these tokens are generated in a diffusion manner. When considering the modeling of long sequences, as subsequent keyframes are produced based on all preceding text and image information, this can also be viewed as a form of temporal auto-regressive modeling.
+
+We have tried to train Show-o using instructional examples and present qualitative examples in Fig. 8 (b). For example, given a question "Can you guide me through making Avocado and Apple Juice", Show-o exhibits the capability to generate video keyframes with text descriptions related to the question. It is apparent that the generated keyframes are temporally consistent.
+
+
+
+
+
+
+
+
+Figure 7: Images generated by Show-o. Text prompts are provided in Appendix G.
+
+
+
+
+
+# 4.5VIDEO UNDERSTANDING AND GENERATION
+
+Beyond image understanding and generation, we have explored Show-o to support video understanding and generation. We inflate the existing pre-trained MAGVIT-v2 to support encoding videos into discrete tokens, compressing an 8 FPS video tensor of $3 \times 17 \times 256 \times 256$ into $5 \times 16 \times 16$ . In this way, following the sequence format and omni-attention of multimodal understanding and generation introduced in Section 3.2, it is convenient to fine-tune the existing Show-o to involve video understanding and generation. One can observe visual examples in Figs. 8 (c) and (d) that Show-o can accurately comprehend and describe the variations in the video and generate consistent video frames with "a jeep car is approaching from a distance". More examples can be found in Appendix H.
+
+# 4.6 ABLATION STUDIES
+
+Impact of dataset scale and image resolution on multimodal understanding. As only discrete image tokens are extracted from the vision tokenizer, it is required to learn image token
+
+Table 4: Impact of dataset scale and image resolution on the learning of discrete image token embeddings for multimodal understanding.
+
+