Krisanu Sarkar

Krisanu Sarkar

Current synthetic data pipelines for computer vision generate images without diagnosing what the downstream model actually needs. This open-loop paradigm treats synthetic data as cheap real data, randomly sampling the generator's output space and hoping to cover the model's failure modes. We argue this fundamentally misuses synthetic data's unique property: the controllable, independent variation of scene factors.Drawing on the statistical theory of Design of Experiments (DoE), we propose Synthetic Designed Experiments for Representational Sufficiency (SDRS). SDRS treats the downstream model as a black-box system and the synthetic generator as an experimental apparatus. Using fractional factorial designs, SDRS efficiently audits a model's factor-sensitivity profile via ANOVA decomposition. It classifies failures into two actionable types: Type I gaps (coverage failures on underrepresented factor levels) and Type II gaps (reliance on spurious nuisance dependencies). The audit then prescribes targeted synthetic data to address each gap type. We validate SDRS on three experiments: (1) a controlled diagnostic on dSprites with planted biases, where the audit correctly identifies both gap types and targeted data improves accuracy from 49.9% to 79.0%; (2) a dense segmentation task on procedural scenes, where detecting background-complexity shortcuts and applying targeted data improves mIoU from 0.948 to 0.998; and (3) an entanglement detection experiment showing that the ANOVA audit identifies cross-factor contamination in imperfect generators. Finally, we show that per-factor invariance penalties can transfer sensitivity between factors, identifying an open problem for representation-level correction.

Krisanu Sarkar

We analyze the score field of a diffusion generative model through a Burgers-type evolution law. For VE diffusion, the heat-evolved data density implies that the score obeys viscous Burgers in one dimension and the corresponding irrotational vector Burgers system in $\R^d$, giving a PDE view of \emph{speciation transitions} as the sharpening of inter-mode interfaces. For any binary decomposition of the noised density into two positive heat solutions, the score separates into a smooth background and a universal $\tanh$ interfacial term determined by the component log-ratio; near a regular binary mode boundary this yields a normal criterion for speciation. In symmetric binary Gaussian mixtures, the criterion recovers the critical diffusion time detected by the midpoint derivative of the score and agrees with the spectral criterion of Biroli, Bonnaire, de~Bortoli, and Mézard (2024). After subtracting the background drift, the inter-mode layer has a local Burgers $\tanh$ profile, which becomes global in the symmetric Gaussian case with width $σ_τ^2/a$. We also quantify exponential amplification of score errors across this layer, show that Burgers dynamics preserves irrotationality, and use a change of variables to reduce the VP-SDE to the VE case, yielding a closed-form VP speciation time. Gaussian-mixture formulas are verified to machine precision, and the local theorem is checked numerically on a quartic double-well.

Krisanu Sarkar

We study cross-modal alignment between independently pretrained vision (DINOv2) and language (all-MiniLM-L6-v2) encoders using the functional map framework from computational geometry, which represents correspondence between representation manifolds as a compact linear operator between graph Laplacian eigenbases. While the framework underperforms Procrustes alignment and relative representations for cross-modal retrieval across all supervision budgets, it reveals a structural property of multimodal representations. We find that the Laplacian eigenvalue spectra of the two encoders are quantitatively similar (normalized spectral distance 0.043), indicating that independently trained models develop manifolds of comparable intrinsic complexity. However, the functional map exhibits near-zero diagonal dominance (mean below 0.05) and large orthogonality error (70.15), showing that the eigenvector bases are effectively unaligned. We term this decoupling the spectral complexity--orientation gap: models converge in how much structure they capture but not in how they organize it. This gap defines a boundary condition for spectral alignment methods and motivates three diagnostic quantities : diagonal dominance, orthogonality deviation, and Laplacian commutativity error for characterizing cross-modal representation compatibility.

Krisanu Sarkar

The persistent challenge of catastrophic forgetting in neural networks has motivated extensive research in continual learning . This work presents a novel continual learning framework that integrates Fisher-weighted asymmetric regularization of parameter variances within a variational learning paradigm. Our method dynamically modulates regularization intensity according to parameter uncertainty, achieving enhanced stability and performance. Comprehensive evaluations on standard continual learning benchmarks including SplitMNIST, PermutedMNIST, and SplitFashionMNIST demonstrate substantial improvements over existing approaches such as Variational Continual Learning and Elastic Weight Consolidation . The asymmetric variance penalty mechanism proves particularly effective in maintaining knowledge across sequential tasks while improving model accuracy. Experimental results show our approach not only boosts immediate task performance but also significantly mitigates knowledge degradation over time, effectively addressing the fundamental challenge of catastrophic forgetting in neural networks

Krisanu Sarkar

In this work, we develop new generalization bounds for neural networks trained on data supported on Riemannian manifolds. Existing generalization theories often rely on complexity measures derived from Euclidean geometry, which fail to account for the intrinsic structure of non-Euclidean spaces. Our analysis introduces a geometric refinement: we derive covering number bounds that explicitly incorporate manifold-specific properties such as sectional curvature, volume growth, and injectivity radius. These geometric corrections lead to sharper Rademacher complexity bounds for classes of Lipschitz neural networks defined on compact manifolds. The resulting generalization guarantees recover standard Euclidean results when curvature is zero but improve substantially in settings where the data lies on curved, low-dimensional manifolds embedded in high-dimensional ambient spaces. We illustrate the tightness of our bounds in negatively curved spaces, where the exponential volume growth leads to provably higher complexity, and in positively curved spaces, where the curvature acts as a regularizing factor. This framework provides a principled understanding of how intrinsic geometry affects learning capacity, offering both theoretical insight and practical implications for deep learning on structured data domains.

Krisanu Sarkar

We introduce Hindsight-Guided Momentum (HGM), a first-order optimization algorithm that adaptively scales learning rates based on the directional consistency of recent updates. Traditional adaptive methods, such as Adam or RMSprop , adapt learning dynamics using only the magnitude of gradients, often overlooking important geometric cues.Geometric cues refer to directional information, such as the alignment between current gradients and past updates, which reflects the local curvature and consistency of the optimization path. HGM addresses this by incorporating a hindsight mechanism that evaluates the cosine similarity between the current gradient and accumulated momentum. This allows it to distinguish between coherent and conflicting gradient directions, increasing the learning rate when updates align and reducing it in regions of oscillation or noise. The result is a more responsive optimizer that accelerates convergence in smooth regions of the loss surface while maintaining stability in sharper or more erratic areas. Despite this added adaptability, the method preserves the computational and memory efficiency of existing optimizers.By more intelligently responding to the structure of the optimization landscape, HGM provides a simple yet effective improvement over existing approaches, particularly in non-convex settings like that of deep neural network training.