Paper deep dive

OOD Generalization as a Bifurcation Problem

Nguyen-Thanh-Luong Doan, Quang-Vu Nguyen, Tang-Phu-Quy Le, Cong-Phap Huynh

Year: 2026Venue: arXiv preprintArea: cs.LGType: PreprintEmbeddings: 30

Intelligence

Status: not_run | Model: - | Prompt: - | Confidence: 0%

Entities (0)

No extracted entities yet.

Relation Signals (0)

No relation signals yet.

Cypher Suggestions (0)

No Cypher suggestions yet.

Abstract

Abstract:Systematic out-of-distribution (OOD) generation remains a critical bottleneck for continuous-time generative models. While standard joint classifier-free guidance (CFG) routinely fails to synthesize unobserved concept combinations, exact decomposed scoring generalizes robustly at the cost of severe computational overhead. In this work, we reveal that compositional binding is not a uniform process but a highly localized phase transition. We identify the semantic bifurcation window - the precise temporal interval where joint and decomposed vector fields meaningfully diverge. Exploiting this dynamic, we propose surgical guidance, a hybrid sampling strategy that restricts exact multi-pass scoring strictly to this critical window. On an OOD bi-digit MNIST testbed, surgical guidance achieves state-of-the-art compositional fidelity at a fraction of the inference cost, yielding a +5.3% absolute improvement in pairwise accuracy over the joint baseline by intervening during just the first 15% of the diffusion trajectory. Furthermore, our empirical analysis uncovers a fundamental topological divide: diffusion models (SDEs) force conceptual resolution immediately at peak noise, whereas Conditional Flow Matching (ODEs) delays structural binding until intermediate features emerge, establishing a new temporal framework for accelerating large-scale generative decoding.

PDF

Open source PDF →Open local PDF →

Trouble viewing inline? Open PDF directly →

Full Text

26,224 characters extracted from source content.

Expand or collapse full text

OOD Generalization as a Bifurcation Problem Nguyen-Thanh-Luong Doan Affiliation: The University of Danang - Vietnam-Korea University of Information and Communication Technology E-mail dntluong, nqvu, quyltp.22git, hcphap@vku.udn.vn Quang-Vu Nguyen,✉ †thanks: ✉˜Corresponding author Affiliation: The University of Danang - Vietnam-Korea University of Information and Communication Technology E-mail dntluong, nqvu, quyltp.22git, hcphap@vku.udn.vn Tang-Phu-Quy Le Affiliation: The University of Danang - Vietnam-Korea University of Information and Communication Technology E-mail dntluong, nqvu, quyltp.22git, hcphap@vku.udn.vn and Cong-Phap Huynh Affiliation: The University of Danang - Vietnam-Korea University of Information and Communication Technology E-mail dntluong, nqvu, quyltp.22git, hcphap@vku.udn.vn Abstract Systematic out-of-distribution (OOD) generation remains a critical bottleneck for continuous-time generative models. While standard joint classifier-free guidance (CFG) routinely fails to synthesize unobserved concept combinations, exact decomposed scoring generalizes robustly at the cost of severe computational overhead. In this work, we reveal that compositional binding is not a uniform process but a highly localized phase transition. We identify the “semantic bifurcation window”—the precise temporal interval where joint and decomposed vector fields meaningfully diverge. Exploiting this dynamic, we propose surgical guidance, a hybrid sampling strategy that restricts exact multi-pass scoring strictly to this critical window. On an OOD bi-digit MNIST testbed, surgical guidance achieves state-of-the-art compositional fidelity at a fraction of the inference cost, yielding a +5.3%+5.3\% absolute improvement in pairwise accuracy over the joint baseline by intervening during just the first 15%15\% of the diffusion trajectory. Furthermore, our empirical analysis uncovers a fundamental topological divide: diffusion models (SDEs) force conceptual resolution immediately at peak noise, whereas Conditional Flow Matching (ODEs) delays structural binding until intermediate features emerge, establishing a new temporal framework for accelerating large-scale generative decoding. Keywords: out-of-distribution generalization compositional generation diffusion models conditional flow matching surgical guidance classifier-free guidance. 1 Introduction Systematic compositional generalization is a fundamental challenge for deep generative models [4, 9, 16, 12]. While continuous-time models like diffusion and Conditional Flow Matching (CFM) [14] exhibit remarkable synthesis capabilities, they consistently struggle to generate out-of-distribution (OOD) concept combinations unseen during training. Standard joint classifier-free guidance (CFG) [11] models in-distribution data well but fails to disentangle and recombine independent features. Conversely, decomposed scoring paradigms [5, 15]—rooted in the product-of-experts formulation of Energy-Based Models and Composable Diffusion—achieve robust OOD generalization. However, this explicit mathematical factorization demands multiple forward network passes per integration step, introducing severe computational overhead. To systematically investigate this inference trade-off, we establish a controlled bi-digit MNIST [13] testbed. We train models to generate paired digits from a heavily restricted subset of observed combinations. True generalization requires the model to correctly synthesize novel, unobserved digit pairs strictly by recombining disentangled left and right positional embeddings (Figure 1). This isolated environment allows us to track the exact dynamics of compositional binding. Observed Training Pairsd1=2d_1\!=\!2d1=3d_1\!=\!3d1=5d_1\!=\!5d1=8d_1\!=\!82 32 53 53 85 85 28 28 3DisentangledEmbeddingsEmbLEmb_L2358× ×recombineEmbREmb_R2358c=EmbL(d1)|EmbR(d2)c=Emb_L(d_1)\,\|\,Emb_R(d_2)each digit learnedindependently per positionNovel OOD Pairs2 83 25 38 5unseen during training Figure 1: Bi-digit MNIST out-of-distribution (OOD) generalization testbed. By analyzing the vector fields of both SDE (Diffusion) and ODE (CFM) trajectories, we demonstrate that joint and decomposed scores do not conflict uniformly across time. Instead, they exhibit a localized phase transition. We formalize this temporal region as the semantic bifurcation window, defining the precise interval where the generative process actively resolves compositional conflicts. Exploiting this dynamical behavior, we propose surgical guidance. This dynamic sampling strategy dynamically switches objective functions during generation, applying exact decomposed scoring strictly within the bifurcation window while utilizing computationally efficient joint CFG everywhere else. Surgical guidance matches the OOD performance of full decomposed generation at a fraction of the inference cost. Furthermore, our empirical evaluations expose a distinct topological divergence between generative frameworks: diffusion models force compositional binding almost immediately at peak noise, whereas flow matching delays structural resolution until intermediate features emerge. These results prove that conceptual attributes are untied and bound at distinct, predictable temporal phases, unlocking targeted acceleration strategies for large-scale generative architectures. 2 Background At the heart of our investigation lies a fundamental tension between how generative models learn distributions and how we expect them to compose concepts. Historically, compositionality was treated as a static architectural constraint or a latent space entanglement problem. However, recent paradigms in Energy-Based Models (EBMs) [10, 17, 7, 6] and continuous-time generative frameworks [18, 15] suggest that composition is inherently probabilistic and dynamic. The theoretical foundation for our exact compositional scoring originates from the product-of-experts formulation in EBMs. To combine independent concepts—such as a left digit L and a right digit R—we assume their joint probability factorizes proportionally to their individual marginals over the unconditional prior: p⁡(x∣L,R)∝p⁡(x)p⁡(x∣L)p⁡(x)p⁡(x∣R)p⁡(x)p(x L,R) p(x) p(x L)p(x) p(x R)p(x) (1) In the energy domain, where a distribution is defined by p⁡(x)∝e−E⁡(x)p(x) e^-E(x), this relationship elegantly collapses into a simple linear addition of their respective energy functions. Composable Diffusion [15] directly maps this static EBM logic onto the dynamic generative trajectory. Because the score network functionally parameterizes the gradient of the data distribution’s log-density, the EBM product rule dictates the update step. For a generic vector field vθ(zt,t,c)v_θ(z_t,t,c)—which represents either the noise prediction ϵθ _θ in diffusion SDEs or the velocity field in Conditional Flow Matching ODEs—standard classifier-free guidance (CFG) extrapolates away from the unconditional prediction ∅ scaled by a guidance weight ω: v~θ(zt,t,c)=vθ(zt,t,∅)+ω⁡(vθ(zt,t,c)−vθ(zt,t,∅)) v_θ(z_t,t,c)=v_θ(z_t,t, )+ω (v_θ(z_t,t,c)-v_θ(z_t,t, ) ) (2) When a model is tasked with generating a multi-concept image, the conventional strategy relies on a joint score. The network is conditioned on the concatenated concepts simultaneously, relying entirely on the assumption that the model has learned a perfectly disentangled representation during training: v~joint=vθ(zt,t,∅)+ω⁡(vθ(zt,t,L,R)−vθ(zt,t,∅)) v_joint=v_θ(z_t,t, )+ω (v_θ(z_t,t,L,R)-v_θ(z_t,t, ) ) (3) While computationally efficient (requiring only two forward passes per step), joint scoring frequently collapses when faced with out-of-distribution combinations. The mathematically rigorous alternative is the decomposed score. By explicitly enforcing the log-gradient factorization derived from Composable Diffusion, we query the network independently for the left and right concepts before assembling their guidance vectors: v~decomp=vθ(zt,t,∅)+ω⁡(vθ(zt,t,L,∅)+vθ(zt,t,∅,R)−2vθ(zt,t,∅)) v_decomp=v_θ(z_t,t, )+ω (v_θ(z_t,t,L, )+v_θ(z_t,t, ,R)-2v_θ(z_t,t, ) ) (4) This explicit factorization is highly reliable for OOD generalization, yet the computational penalty is severe. Evaluating the decomposed score demands three distinct forward passes per integration step. Reflecting on this overhead, we hypothesized that this rigid, continuous mathematical enforcement is largely redundant. It is highly improbable that the network actively binds independent concepts uniformly across the entire generation trajectory; rather, there must exist a specific, critical temporal phase where these independent vector fields actually interfere, interact, and resolve. 3 Semantic Bifurcation Window and Surgical Guidance We questioned whether the model actively struggles with compositional binding throughout the entire generation process. If the network only negotiates conceptual conflicts during a specific phase, enforcing a rigorous and computationally expensive decomposed score at every integration step is inherently wasteful. To investigate this, we quantified the functional discrepancy—which we term interference—between the joint and decomposed vector fields. By tracking the Mean Squared Error (MSE) and Cosine Similarity cosS_cos (v~joint,v~decomp)( v_joint, v_decomp) across the normalized sampling trajectory, a striking dynamic pattern emerged. Figure 2: Divergence between joint and decomposed scoring vector fields. The two scoring strategies dictate nearly identical trajectories at the extremes of the generation process. Early on, the latent state is dominated by structureless noise; late in the process, the topology is already rigidly established. However, the vector fields diverge drastically during a highly localized intermediate phase. We formalize this critical region of instability as the semantic bifurcation window, denoted by the interval [tstart,tend][t_start,t_end]. It is exclusively within this temporal pocket that the network is forced to untangle and resolve the conflicting spatial gradients of the independent digit embeddings. Figure 3: Visualizing attractor collapse during the semantic bifurcation window. Table 1: Out-of-distribution compositional generalization errors. Values represent the number of misclassified samples out of 300 novel held-out pairs (lower is better). Method (2,8) (3,2) (5,3) (8,5) (2,2) (3,3) (5,5) (8,8) Total CFDG (joint) 74 47 45 57 32 12 50 6 323 CFDG (decomposed) 27 28 29 30 15 3 37 2 171 CFM (joint) 19 23 28 21 14 14 18 21 158 CFM (decomposed) 18 21 9 14 8 2 15 12 99 This observation fundamentally alters how we can approach compositional sampling. If the standard joint score naturally approximates the decomposed score outside of this active window, we can bypass the three-pass mathematical factorization entirely for the majority of the trajectory. This led us to formulate surgical guidance, a hybrid sampling strategy that dynamically switches the guidance objective based on the current timestep: v~surgical(zt,t)=v~decomp(zt,t)if t∈[tstart,tend]v~joint(zt,t)otherwise v_surgical(z_t,t)= cases v_decomp(z_t,t)&if t∈[t_start,t_end]\\ v_joint(z_t,t)&otherwise cases (5) By restricting the exact decomposed evaluation strictly to the semantic bifurcation window, surgical guidance effectively insulates the generative process against out-of-distribution collapse. This targeted intervention guarantees the structural integrity of novel concept combinations while preserving the computational efficiency of standard classifier-free guidance everywhere else. CFMz0∼z_0\! \!Nz1=xz_1\!=\!xt=0t\!=\!0t=1t\!=\!1Forward EulerJointSurgicalJointv^=v∅+w⁡(vLR−v∅) v=v_ +w(v_LR-v_ )v^=v∅+w⁡(vL∅+v∅R−2v∅) v=v_ +w(v_L \!+\!v_ R\!-\!2v_ )v^=v∅+w⁡(vLR−v∅) v=v_ +w(v_LR-v_ )DDIMz0=xz_0\!=\!xzT∼z_T\! \!Nt=0t\!=\!0t=1t\!=\!1DDIM backwardJointSurgicalJointϵ^=ϵ∅+w⁡(ϵLR−ϵ∅) ε= _ +w( _LR- _ )ϵ^=ϵ∅+w⁡(ϵL∅+ϵ∅R−2ϵ∅) ε= _ +w( _L \!+\! _ R\!-\!2 _ )ϵ^=ϵ∅+w⁡(ϵLR−ϵ∅) ε= _ +w( _LR- _ )Semantic Bifurcation Windowp=0.3p\!=\!0.3p=0.7p\!=\!0.7sampling progress p2 fwd/step3 fwd/step2 fwd/step3 fwd/stepJoint guidance: e∅+w⁡(eLR−e∅)e_ +w(e_LR-e_ )Surgical guidance: e∅+w⁡(eL∅+e∅R−2e∅)e_ +w(e_L +e_ R-2e_ ) Figure 4: Surgical guidance mechanism across CFM and DDIM frameworks. Algorithm 1 Surgical Guidance for Conditional Flow Matching (CFM) 1: Initial noise z0∼⁡(0,I)z_0 (0,I), concepts (cL,cR)(c_L,c_R), bifurcation window [τstart,τend][ _start, _end], integration steps N, guidance scale ω 2: Generated image x∗x^* 3: Δt←1/N t← 1/N 4: z←z0z← z_0 5: for i=0i=0 to N−1N-1 do 6: t←i⋅Δt← i· t 7: prog←i/Nprog← i/N 8: vuncond←vθ(z,t,∅,∅)v_uncond← v_θ(z,t, , ) 9: if τstart≤prog≤τend _start ≤ _end then ⊳ Semantic Bifurcation Window (3 NFEs) 10: vleft←vθ(z,t,cL,∅)v_left← v_θ(z,t,c_L, ) 11: vright←vθ(z,t,∅,cR)v_right← v_θ(z,t, ,c_R) 12: v~t←vuncond+ω⋅(vleft+vright−2⋅vuncond) v_t← v_uncond+ω·(v_left+v_right-2· v_uncond) 13: else⊳ Standard Joint CFG (2 NFEs) 14: vcond←vθ(z,t,cL,cR)v_cond← v_θ(z,t,c_L,c_R) 15: v~t←vuncond+ω⋅(vcond−vuncond) v_t← v_uncond+ω·(v_cond-v_uncond) 16: end if 17: z←z+v~t⋅Δtz← z+ v_t· t ⊳ Forward Euler ODE step 18: end for 19: x∗←Decoder(z)x^* (z) 20: return x∗x^* Algorithm 2 Surgical Guidance for Diffusion Models (DDIM) 1: Initial noise zN∼⁡(0,I)z_N (0,I), concepts (cL,cR)(c_L,c_R), bifurcation window [τstart,τend][ _start, _end], integration steps N, guidance scale ω 2: Generated image x∗x^* 3: Δt←1/N t← 1/N 4: z←zNz← z_N 5: for i=0i=0 to N−1N-1 do 6: t←0.999−i⋅Δt← 0.999-i· t 7: tnext←t−Δt_next← t- t 8: prog←i/Nprog← i/N 9: ϵuncond←ϵθ(z,t,∅,∅) _uncond← _θ(z,t, , ) 10: if τstart≤prog≤τend _start ≤ _end then ⊳ Semantic Bifurcation Window (3 NFEs) 11: ϵleft←ϵθ(z,t,cL,∅) _left← _θ(z,t,c_L, ) 12: ϵright←ϵθ(z,t,∅,cR) _right← _θ(z,t, ,c_R) 13: ϵ~t←ϵuncond+ω⋅(ϵleft+ϵright−2⋅ϵuncond) ε_t← _uncond+ω·( _left+ _right-2· _uncond) 14: else⊳ Standard Joint CFG (2 NFEs) 15: ϵcond←ϵθ(z,t,cL,cR) _cond← _θ(z,t,c_L,c_R) 16: ϵ~t←ϵuncond+ω⋅(ϵcond−ϵuncond) ε_t← _uncond+ω·( _cond- _uncond) 17: end if 18: z0pred←z−1−α¯t⋅ϵ~tα¯tz_0^pred← z- 1- α_t· ε_t α_t 19: z←α¯tnext⋅z0pred+1−α¯tnext⋅ϵ~tz← α_t_next· z_0^pred+ 1- α_t_next· ε_t ⊳ DDIM step 20: end for 21: x∗←Decoder(z)x^* (z) 22: return x∗x^* 4 Experiments 4.1 Experimental Setup and Evaluation Metrics To empirically validate surgical guidance and locate the exact semantic bifurcation windows, we evaluate out-of-distribution generation on our controlled bi-digit MNIST testbed. We systematically sweep a sliding intervention window of fixed lengths (l=0.1l=0.1 and l=0.15l=0.15) across the normalized integration timeline. Within each designated window, we apply the exact three-pass decomposed score; outside of it, we revert to the standard two-pass joint score. To rigorously quantify compositional failures, we employ a disentangled classifier that independently predicts the left and right digits of a generated image. A sample is strictly counted as a compositional error (Table 1) if either spatial prediction diverges from the conditioning label. Furthermore, to measure generative fidelity, we compute the Fréchet Inception Distance (FID) [8] for surgical sampling on the out-of-distribution pairs (Table 2). Features for the FID calculation are extracted from the penultimate layer of our disentangled digit classifier, yielding a 128-dimensional representation tailored to the spatial halves. Lower FID values indicate stronger distributional alignment with the ground-truth training set. 4.2 Empirical Results and Topological Divergence Our evaluation measures the absolute improvement in OOD pairwise accuracy over the joint CFG baseline across 50 random seeds. The empirical results expose a stark topological divide between generative frameworks. For diffusion models (SDEs), the joint baseline achieves an 87.2%87.2\% pairwise accuracy. As illustrated in Figures 5 and 6, the optimal semantic bifurcation window occurs aggressively early in the generation process. Applying surgical guidance strictly during the initial 10%10\% to 15%15\% of the trajectory (progress ∈[0.00,0.15]∈[0.00,0.15]) yields a massive +5.3%+5.3\% absolute improvement. This indicates that the SDE framework forces compositional binding almost immediately out of pure noise, locking in structural geometry well before any high-frequency details are synthesized. Conversely, Conditional Flow Matching (CFM) dictates a fundamentally different dynamic. CFM inherently possesses a stronger joint baseline of 94.2%94.2\%. More importantly, its bifurcation window is temporally delayed. Figures 7 and 8 demonstrate that the optimal intervention zone peaks between progress bounds [0.10,0.25][0.10,0.25], delivering up to a +1.1%+1.1\% improvement. The ODE trajectory effectively defers structural composition. Rather than resolving conceptual conflicts at peak noise, flow matching waits until the latent state has partially smoothed into intermediate features before untangling the left and right positional embeddings. Our quantitative FID analysis (Table 2) reinforces these observations. CFDG surgical sampling attains a lower mean FID than CFM across all held-out compositions (15.415.4 vs. 16.316.3), corroborating the error-count analysis. Interestingly, we observe an architectural quirk: cross-digit pairs consistently yield lower FIDs than repeated-digit compositions (e.g., (k,k)(k,k) pairs). This suggests that our localized surgical intervention is inherently more effective at disentangling semantically distinct digits than isolating identical, overlapping concepts. Table 2: Fréchet Inception Distance (FID) for surgical sampling on out-of-distribution pairs. Lower values indicate stronger distributional alignment with the training set. Method (2,8) (3,2) (5,3) (8,5) (2,2) (3,3) (5,5) (8,8) Mean CFDG (surgical) 16.9 14.5 13.5 13.9 10.5 18.7 11.8 23.5 15.4 CFM (surgical) 16.9 15.7 15.5 16.4 12.7 19.0 14.3 19.6 16.3 Figure 5: Diffusion bifurcation sweep (l=0.1l=0.1). Optimal intervention occurs immediately at peak noise, yielding a +5.1%+5.1\% improvement. Figure 6: Diffusion bifurcation sweep (l=0.15l=0.15). Expanding the window slightly captures a +5.3%+5.3\% gain, confirming early topological resolution. Figure 7: CFM bifurcation sweep (l=0.1l=0.1). The semantic window is delayed, peaking at [0.15,0.25][0.15,0.25] to resolve compositional conflicts. Figure 8: CFM bifurcation sweep (l=0.15l=0.15). Optimal structural binding confirms the delayed ODE resolution phase. 5 Conclusion and Future Potential The identification of the semantic bifurcation window shifts the paradigm of compositional generation from a spatial representation problem to a dynamical systems problem. By isolating the exact temporal phase where generative models resolve conceptual interference, surgical guidance demonstrates that mathematically rigorous, multi-pass scoring is only necessary for a brief fraction of the sampling trajectory. This targeted intervention successfully preserves the out-of-distribution fidelity of full decomposed scoring while bypassing redundant neural evaluations elsewhere. Looking forward, the implications of temporal structural binding extend far beyond controlled testbeds. Modern large-scale text-to-image architectures heavily rely on computationally expensive guidance mechanisms to enforce complex prompt adherence. Mapping the bifurcation windows of massive latent diffusion and flow-based models [19, 2, 1, 3] could unlock highly optimized dynamic decoding algorithms. By anticipating exactly when a network binds independent semantic concepts, we can systematically accelerate inference pipelines without sacrificing the structural integrity of multi-concept synthesis. Acknowledgement This research is funded by Vietnam - Korea University of Information and Communication Technology under project number ÐHVH-2026-21. References [1] Black Forest Labs and Stephen Batifol and Andreas Blattmann and Frederic Boesel and Saksham Consul and Cyril Diagne and Tim Dockhorn and Jack English and Zion English and Patrick Esser and Sumith Kulal and Kyle Lacey and Yam Levi and Cheng Li and Dominik Lorenz and Jonas Müller and Dustin Podell and Robin Rombach and Harry Saini and Axel Sauer and Luke Smith (2025) FLUX.1 kontext: flow matching for in-context image generation and editing in latent space. External Links: 2506.15742, Link Cited by: §5. [2] Black Forest Labs (2024) FLUX. Note: https://github.com/black-forest-labs/flux Cited by: §5. [3] Black Forest Labs (2025) FLUX.2: Frontier Visual Intelligence. Note: https://bfl.ai/blog/flux-2 Cited by: §5. [4] N. Chomsky (1965) Aspects of the theory of syntax. MIT Press, Cambridge, MA. Cited by: §1. [5] Y. Du, S. Li, and I. Mordatch (2020) Compositional visual generation with energy based models. Advances in Neural Information Processing Systems 33, p. 6637–6647. Cited by: §1. [6] Y. Du, S. Li, Y. Sharma, J. Tenenbaum, and I. Mordatch (2021) Unsupervised learning of compositional energy concepts. Advances in Neural Information Processing Systems 34, p. 15608–15620. Cited by: §2. [7] K. Greff, R. L. Kaufman, R. Kabra, N. Watters, C. Burgess, D. Zoran, L. Matthey, M. Botvinick, and A. Lerchner (2019) Multi-object representation learning with iterative variational inference. In International conference on machine learning, p. 2424–2433. Cited by: §2. [8] M. Heusel, H. Ramsauer, T. Unterthiner, B. Nessler, and S. Hochreiter (2017) Gans trained by a two time-scale update rule converge to a local nash equilibrium. Advances in neural information processing systems 30. Cited by: §4.1. [9] I. Higgins, L. Matthey, A. Pal, C. Burgess, X. Glorot, M. Botvinick, S. Mohamed, and A. Lerchner (2017) Beta-VAE: learning basic visual concepts with a constrained variational framework. In International Conference on Learning Representations, External Links: Link Cited by: §1. [10] G. E. Hinton (2002) Training products of experts by minimizing contrastive divergence. Neural computation 14 (8), p. 1771–1800. Cited by: §2. [11] J. Ho and T. Salimans (2021) Classifier-free diffusion guidance. In NeurIPS 2021 Workshop on Deep Generative Models and Downstream Applications, External Links: Link Cited by: §1. [12] R. Jacob, M. Sam, and D. Williams (forthcoming) From frege to chatgpt: compositionality in language, cognition, and deep neural networks. In Neuroscience and Philosophy. Vol. 2, de Brigard Felipe and S. Walter (Eds.), Cited by: §1. [13] Y. LeCun, C. Cortes, and C. Burges (2010) MNIST handwritten digit database. ATT Labs [Online]. Available: http://yann.lecun.com/exdb/mnist 2. Cited by: §1. [14] Y. Lipman, R. T. Q. Chen, H. Ben-Hamu, M. Nickel, and M. Le (2023) Flow matching for generative modeling. In The Eleventh International Conference on Learning Representations, External Links: Link Cited by: §1. [15] N. Liu, S. Li, Y. Du, A. Torralba, and J. B. Tenenbaum (2022) Compositional visual generation with composable diffusion models. In European conference on computer vision, p. 423–439. Cited by: §1, §2, §2. [16] G. Marcus, E. Davis, and S. Aaronson (2022) A very preliminary analysis of dall-e 2. arXiv preprint arXiv:2204.13807. Cited by: §1. [17] N. Shazeer, *. Mirhoseini, *. Maziarz, A. Davis, Q. Le, G. Hinton, and J. Dean (2017) Outrageously large neural networks: the sparsely-gated mixture-of-experts layer. In International Conference on Learning Representations, External Links: Link Cited by: §2. [18] J. Song, C. Meng, and S. Ermon (2021) Denoising diffusion implicit models. In International Conference on Learning Representations, External Links: Link Cited by: §2. [19] Y. Song, P. Dhariwal, M. Chen, and I. Sutskever (2023) Consistency models. In Proceedings of the 40th International Conference on Machine Learning, A. Krause, E. Brunskill, K. Cho, B. Engelhardt, S. Sabato, and J. Scarlett (Eds.), Proceedings of Machine Learning Research, Vol. 202, p. 32211–32252. External Links: Link Cited by: §5.

Instant research discovery