Representation Learning via Composite Subspace Flow Matching

José Afonso; Mário Figueiredo

cs.CVOct 23, 2026

Representation Learning via Composite Subspace Flow Matching

Authors: José Afonso, Mário Figueiredo

Organizations: Instituto Superior Técnico, Lisboa, Portugal · Instituto de Telecomunicações, Instituto Superior Técnico, Lisboa, Portugal

Abstract

Distilling semantic meaning from raw image data remains a challenge in machine learning. Current self-supervised paradigms face a trade-off: joint-embedding architectures produce relevant features by aggressively discarding intra-class variance, irreversibly spatial and reconstruction. Conversely, generative models preserve spatial details but struggle to isolate semantics, as their optimization is dominated by pixel variations. To bridge this divide, we propose composite subspace flow matching (CSFM). Rather than corrupting data with unstructured isotropic noise, CSFM formulates representation learning as the variance-minimizing inversion of a deterministic degradation curriculum across spectral, color, and illumination subspaces. By bounding trajectory variance with hard temporal style truncation, a capacity-constrained encoder is driven to ignore transient style artifacts and encode persistent content. Empirical evaluations across CIFAR-10, CIFAR-100, and STL-10 show that CSFM yields competitive discriminative transferability while preserving generative synthesis across both convolutional and vision transformer architectures. By avoiding the collapse pathologies of non-generative methods, CSFM forms a smooth latent space that outperforms standard optimal transport flows on linear probing and few-shot classification, while enabling coherent synthesis and trajectory interpolation. These results confirm that steering the physical dynamics of generative flows offers a scalable paradigm for learning representations that are simultaneously linearly separable and generatively invertible.

Figures & tables

Paradigm	Model	Linear Probe Accuracy			$k$ -NN	Silhouette
Paradigm	Model	100%	10%	1%	$k$ -NN	Silhouette
Pixel Recon.	Autoencoder (AE)	58.07%	51.08%	37.45%	48.37%	-0.0780
Pixel Recon.	CDAE (CSFM Path)	57.27%	50.68%	36.06%	48.81%	-0.0599
Flow Matching	OT-Flow	67.11%	62.25%	45.75%	52.97%	-0.0454
	CSFM ( $t_{1}=1.0$ , No Trunc.)	69.59%	63.33%	46.31%	64.14%	-0.0098
	CSFM ( $t_{1}=0.5$ , Trunc.)	75.29%	71.94%	58.93%	68.34%	-0.0154

Table 1: Representation benchmark on CIFAR-10 across generative and discriminative paradigms (40K steps, baseline 20.81M tier: WRN-28-4 encoder paired with 14.83M U-Net). Best generative result in bold.

Figure 1: Topological manifold analysis via t-SNE embeddings on CIFAR-10 test set. ( Left ) OT-Flow forms dispersed, overlapping representations. ( Middle ) Truncated CSFM ( $t_{1}=0.5$ ) isolates distinct semantic clusters while preserving global manifold connectivity. ( Right ) LeJEPA shatters representations into isolated topological islands.

Model	Steps	Linear (100%)	1% Few-Shot	$k$ -NN	Silhouette
CSFM ( $t_{1}=0.5$ )	40K	77.31%	63.74%	72.08%	0.0068
	80K	78.59%	63.72%	71.77%	-0.0038
	120K	78.51%	63.45%	70.73%	-0.0166
CSFM ( $t_{1}=1.0$ )	40K	69.86%	48.16%	64.54%	-0.0099
	80K	72.19%	49.01%	66.09%	-0.0114
	120K	73.59%	49.34%	66.74%	-0.0154

Table 2: Decoder capacity scaling on CIFAR-10 with a fixed WideResNet-28-4 encoder (5.98M parameters) paired with the high-capacity U-Net regressor (59.10M parameters).

Asymmetric Routing Cutoff ( $t_{0}$ ) with $t_{1}=1.0$
Configuration	Linear Probe	1% Few-Shot	$k$ -NN	Silhouette
$t_{0}=0.0$ (No Routing)	69.09%	46.41%	63.13%	-0.0185
$t_{0}=0.001$	69.12%	46.00%	63.04%	-0.0202
$t_{0}=0.025$	69.22%	47.84%	64.23%	-0.0126
$t_{0}=0.050$	69.52%	47.41%	64.38%	-0.0105
Style Truncation Horizon ( $t_{1}$ ) with $t_{0}=0.05$

Table 3: Ablation of temporal boundaries on CIFAR-10 (20.81M baseline tier, 40K steps).

Benchmark Axis	Dataset	Model Backbone	Steps	Linear Probe	1% Few-Shot	$k$ -NN ( $k=20$ )
Semantic Cardinality	CIFAR-100	OT-Flow (CNN)	40K	42.27%	10.66%	33.63%
Semantic Cardinality	CIFAR-100	CSFM (CNN)	40K	51.58%	17.65%	38.97%
Architectural Bias	CIFAR-10	OT-Flow (ViT/DiT)	40K	51.84%	39.15%	56.81%
	CIFAR-10	OT-Flow (ViT/DiT)	80K	53.27%	41.06%	58.36%
	CIFAR-10	CSFM (ViT/DiT)	40K	53.30%	42.27%	55.14%
	CIFAR-10	CSFM (ViT/DiT)	80K	61.49%	49.69%	62.77%

Table 4: Scalability benchmarks across semantic cardinality (CIFAR-100), backbones (CNN vs. ViT/DiT), and spatial resolution (STL-10). Evaluated under strict parameter parity within each tier. Best results per tier in bold.

Appendix figures & tables10 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 6

Operator Decay Rates
Parameter	Value
Illumination ( $k_{\text{illum}}$ )	$3.250$
Color ( $k_{\text{color}}$ )	$2.925$
Frequency ( $k_{\text{freq}}$ )	$2.632$
Curriculum Horizons and Projectors
Style Truncation Boundary ( $t_{1}$ )	$0.50$

Appendix

Table 5: CSFM degradation curriculum and optimization hyperparameters.

Model	Learning Paradigm	Primary Task Objective ( $\mathcal{L}_{\text{task}}$ )
AE	Unconstrained Reconstruction	$\mathbb{E}_{x_{0}}\left[\big\\|x_{0}-v_{\theta}(\mathbf{0},0,z)\big\\|_{2}^{2}\right]$
CDAE	Single-Step Denoising	$\mathbb{E}_{t,x_{0},x_{1}}\left[\big\\|x_{0}-\hat{x}_{\theta}(x_{t},t,z)\big\\|_{2}^{2}\right]$
OT-Flow	Continuous Linear Flow	$\mathbb{E}_{t,x_{0},x_{1}}\left[\big\\|v_{\theta}(x_{t},t,z)-(x_{1}-x_{0})\big\\|_{2}^{2}\right]$
LeJEPA	Joint Embedding	$\mathbb{E}_{x_{0},v_{1},v_{2}}\left[\frac{1}{2}\big\\|z_{v_{1}}-z_{v_{2}}\big\\|_{2}^{2}\right]$

Appendix

Table 6: Primary task objectives (

\mathcal{L}_{\text{task}}

) for comparative baselines. We denote

x_{1}\sim\mathcal{N}(0,I_{D})

as the spatial prior and

v_{1},v_{2}

as independent stochastic views.

Regularization ( $\lambda_{\text{reg}}$ )	Linear Probing Accuracy			$k$ -NN ( $k=20$ )	Silhouette Score
Regularization ( $\lambda_{\text{reg}}$ )	100% Data	10% Data	1% Data	$k$ -NN ( $k=20$ )	Silhouette Score
$1\times 10^{-3}$	18.62%	18.12%	17.85%	31.13%	-0.1763
$5\times 10^{-3}$	65.18%	59.81%	54.57%	64.41%	-0.0535
$7.5\times 10^{-3}$	71.31%	68.28%	64.21%	70.31%	0.1019
$1\times 10^{-2}$	72.97%	70.26%	66.69%	71.64%	0.1604
$3\times 10^{-2}$	72.49%	71.02%	67.76%	71.65%	0.1738

Appendix

Table 7: Ablation of SIGReg regularization strength (

\lambda_{\text{reg}}

) on the LeJEPA baseline.

Figure 2: Dimensional collapse analysis via singular value decomposition (SVD) on frozen CIFAR-10 test embeddings. ( a ) Normalized singular value spectrum. ( b ) Cumulative explained variance highlighting the 90% energy threshold (

d_{90\%}

Figure 3: Empirical Pearson correlation matrices of the latent bottleneck dimensions on CIFAR-10 test embeddings across OT-Flow ( Left ), truncated CSFM ( $t_{1}=0.5$ , Middle ), and LeJEPA ( Right ).

Model	$\lambda_{\text{reg}}$	Linear Probe	1% Few-Shot	$k$ -NN ( $k=20$ )	Silhouette
CSFM	$0.0$ (Unregularized)	68.98%	45.50%	64.08%	-0.0077
	$\mathbf{1\times 10^{-3}}$	69.09%	46.41%	63.13%	-0.0185
	$5\times 10^{-3}$	68.09%	46.40%	53.85%	-0.0485
	$1\times 10^{-2}$	68.44%	46.31%	53.20%	-0.0221
	$1\times 10^{-1}$	64.61%	44.96%	55.73%	0.0001
OT-Flow	$0.0$ (Unregularized)	66.40%	43.61%	60.40%	-0.0205

Appendix

Table 8: Ablation over structural regularization weight

\lambda_{\text{reg}}

on the 20.81M baseline tier (40K steps). CSFM evaluated with

\kappa=6.0,\beta=0.7,t_{0}=0.0,t_{1}=1.0

Optimal Truncation ( $t_{1}=0.5$ )
Hyperparameter	Linear Probe	1% Few-Shot	$k$ -NN	Silhouette
Base Style Decay Rate ( $\kappa$ ) with $\beta=0.7$
$\kappa=10^{-4}$	75.53%	60.12%	66.33%	0.0062
$\kappa=0.01$	74.80%	58.46%	64.47%	0.0011
$\kappa=0.5$	75.25%	58.72%	65.60%	-0.0047
$\kappa=3.25$	76.10%	58.57%	67.27%	-0.0169

Appendix

Table 9: Ablation of continuous trajectory decay parameters (

\kappa

and

\beta

) on CIFAR-10 (20.81M tier, 40K steps).

Model Configuration	Uncond. $\text{FID}_{1\text{k}}$ ( $\downarrow$ )	Cond. $\text{FID}_{1\text{k}}$ ( $\downarrow$ )	ODE Error ( $N=50$ )	ODE Error ( $N=10$ )
OT-Flow	93.57	119.04	0.00015	0.00327
CSFM ( $t_{1}=1.0$ )	125.97	48.84	0.00505	0.08340
CSFM ( $t_{1}=0.5$ )	95.76	87.88	0.00277	0.04393

Appendix

Table 10: Generative synthesis fidelity (

\text{FID}_{1\text{k}}

) and conditional ODE integration error (MSE vs.

N=1000

baseline) at 120K steps on the high-capacity tier (65.08M total). Best results in bold.

Panel of Figure 4 — Figure 4: Spherical linear interpolation ( $\lambda\in[0,1]$ , green) and extrapolation ( $\lambda\notin[0,1]$ , red) across test-set latent anchors evaluated at 120K steps. Truncated CSFM ( $t_{1}=0.5$ ) demonstrates fluid semantic morphing across intermediate coordinates and maintains structural integrity into the extrapolation regime.

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