Mikhail Osipov

Mikhail Osipov

We present a geometry-driven method for normalizing dysarthric speech by modeling time, frequency, and amplitude distortions as smooth, local Lie group transformations of spectrograms. Scalar fields generate these deformations via exponential maps, and a neural network is trained - using only synthetically warped healthy speech - to infer the fields and apply an approximate inverse at test time. We introduce a spontaneous-symmetry-breaking (SSB) potential that encourages the model to discover non-trivial field configurations. On real pathological speech, the system delivers consistent gains: up to 17 percentage-point WER reduction on challenging TORGO utterances and a 16 percent drop in WER variance, with no degradation on clean CommonVoice data. Character and phoneme error rates improve in parallel, confirming linguistic relevance. Our results demonstrate that geometrically structured warping provides consistent, zero-shot robustness gains for dysarthric ASR.

Mikhail Osipov

We investigate geometric regularization strategies for learned latent representations in encoder--decoder reduced-order models. In a fixed experimental setting for the advection--diffusion--reaction (ADR) equation, we model latent dynamics using a neural ODE and evaluate four regularization approaches applied during autoencoder pre-training: (a) near-isometry regularization of the decoder Jacobian, (b) a stochastic decoder gain penalty based on random directional gains, (c) a second-order directional curvature penalty, and (d) Stiefel projection of the first decoder layer. Across multiple seeds, we find that (a)--(c) often produce latent representations that make subsequent latent-dynamics training with a frozen autoencoder more difficult, especially for long-horizon rollouts, even when they improve local decoder smoothness or related sensitivity proxies. In contrast, (d) consistently improves conditioning-related diagnostics of the learned latent dynamics and tends to yield better rollout performance. We discuss the hypothesis that, in this setting, the downstream impact of latent-geometry mismatch outweighs the benefits of improved decoder smoothness.

Mikhail Osipov

We study the problem of reducing a task cost functional $W : H^s(M) \to \mathbb{R}$, not assumed continuous or differentiable, defined over Sobolev-class signals $S \in H^s(M) $, in the presence of a global symmetry group $G \subset \mathrm{Diff}(M)$. The group acts on signals by pullback, and the cost $W$ is invariant under this action. Such scenarios arise in machine learning and related optimization tasks, where performance metrics may be discontinuous or model-internal. We propose a variational method that exploits the symmetry structure to construct explicit deformations of the input signal. A deformation control field $ φ: M \to \mathbb R^d$, obtained by minimizing an auxiliary energy functional, induces a flow that generically lies in the normal space (with respect to the $L^2$ inner product) to the $G$-orbit of $S$, and hence is a natural candidate to cross the decision boundary of the $G $-invariant cost. We analyze two variants of the coupling term: (1) purely geometric, independent of $W$, and (2) weakly coupled to $W$. Under mild conditions, we show that symmetry-breaking deformations of the signal can reduce the cost. Our approach requires no gradient backpropagation or training labels and operates entirely at test time. It provides a principled tool for optimizing discontinuous invariant cost functionals via Lie-algebraic variational flows.