arXiv subjects

Pengcheng Cheng

Publications and source records attributed to Pengcheng Cheng.

4 recordsLinked to original sources

arXiv2026

Quotient Dynamics, Effective Curvature, and Implicit Bias in Positive Quadratic Networks

Positive quadratic networks admit the low-rank representation f_U(x)=x^top UU^top x, where Uinmathbb{R}^{dtimes r} is identifiable only up to right orthogonal multiplication, representing a rank-r PSD matrix Q=UU^top. We study how this quotient structure governs training dynamics, curvature, recovery, and interpolation bias. On the full-column-rank stratum, we identify mathbb{R}^{dtimes r}_*/O(r) with the rank-r PSD manifold. For smooth objectives L(U)=ell(UU^top), the Euclidean factor gradient is horizontal. Thus, factor gradient flow projects exactly to quotient Riemannian gradient flow, while finite-step gradient descent induces an exact congruence recursion for the predictor. For quadratic regression, we derive the effective Hessian at interpolators as the empirical measurement Gram form restricted to the tangent space relative to the quotient metric. Under Gaussian rank-one measurements, we compute population curvature, prove uniform deviation bounds for the empirical normal operator, construct a spectral initializer, and establish local exponential convergence for gradient flow and linear convergence for small-step descent. Recovery guarantees are explicit but conservative due to reliance on full-space second-moment control. In underdetermined commuting regimes, factor gradient flow becomes an exact entropy mirror flow in joint spectral coordinates. Strictly positive initializations converge to Bregman projections onto the interpolation set. With isotropic initialization q(0)=varepsilon^2mathbf{1}, predictors approach the minimum-trace solution set as varepsilondownarrow0, resolving nonuniqueness via weighted entropy within the invariant joint spectral algebra. Finite-step descent selects interpolants differing from continuous-time Bregman projections by O(eta). Numerical experiments verify these quotient identities, curvature predictions, recovery behaviors, and selection laws.

cs.AI↗

arXiv2026

Quotient Geometry, Effective Curvature, and Implicit Bias in Simple Shallow Neural Networks

Overparameterized shallow neural networks admit substantial parameter redundancy: distinct parameter vectors may represent the same predictor due to hidden-unit permutations, rescalings, and related symmetries. As a result, geometric quantities computed directly in the ambient Euclidean parameter space can reflect artifacts of representation rather than intrinsic properties of the predictor. In this paper, we develop a differential-geometric framework for analyzing simple shallow networks through the quotient space obtained by modding out parameter symmetries on a regular set. We first characterize the symmetry and quotient structure of regular shallow-network parameters and show that the finite-sample realization map induces a natural metric on the quotient manifold. This leads to an effective notion of curvature that removes degeneracy along symmetry orbits and yields a symmetry-reduced Hessian capturing intrinsic local geometry. We then study gradient flows on the quotient and show that only the horizontal component of parameter motion contributes to first-order predictor evolution, while the vertical component corresponds purely to gauge variation. Finally, we formulate an implicit-bias viewpoint at the quotient level, arguing that meaningful complexity should be assigned to predictor classes rather than to individual parameter representatives. Our experiments confirm that ambient flatness is representation-dependent, that local dynamics are better organized by quotient-level curvature summaries, and that in underdetermined regimes, implicit bias is most naturally described in quotient coordinates.

cs.LG↗

arXiv2026

Interpretable Operator Learning for Inverse Problems via Adaptive Spectral Filtering: Convergence and Discretization Invariance

Solving ill-posed inverse problems necessitates effective regularization strategies to stabilize the inversion process against measurement noise. While classical methods like Tikhonov regularization require heuristic parameter tuning, and standard deep learning approaches often lack interpretability and generalization across resolutions, we propose SC-Net (Spectral Correction Network), a novel operator learning framework. SC-Net operates in the spectral domain of the forward operator, learning a pointwise adaptive filter function that reweights spectral coefficients based on the signal-to-noise ratio. We provide a theoretical analysis showing that SC-Net approximates the continuous inverse operator, guaranteeing discretization invariance. Numerical experiments on 1D integral equations demonstrate that SC-Net: (1) achieves the theoretical minimax optimal convergence rate ($O(\delta^{0.5})$ for $s=p=1.5$), matching theoretical lower bounds; (2) learns interpretable sharp-cutoff filters that outperform Oracle Tikhonov regularization; and (3) exhibits zero-shot super-resolution, maintaining stable reconstruction errors ($\approx 0.23$) when trained on coarse grids ($N=256$) and tested on significantly finer grids (up to $N=2048$). The proposed method bridges the gap between rigorous regularization theory and data-driven operator learning.

stat.ML↗

arXiv2026

Gauge-Equivariant Intrinsic Neural Operators for Geometry-Consistent Learning of Elliptic PDE Maps

Learning solution operators of partial differential equations (PDEs) from data has emerged as a promising route to fast surrogate models in multi-query scientific workflows. However, for geometric PDEs whose inputs and outputs transform under changes of local frame (gauge), many existing operator-learning architectures remain representation-dependent, brittle under metric perturbations, and sensitive to discretization changes. We propose Gauge-Equivariant Intrinsic Neural Operators (GINO), a class of neural operators that parameterize elliptic solution maps primarily through intrinsic spectral multipliers acting on geometry-dependent spectra, coupled with gauge-equivariant nonlinearities. This design decouples geometry from learnable functional dependence and enforces consistency under frame transformations. We validate GINO on controlled problems on the flat torus ($\mathbb{T}^2$), where ground-truth resolvent operators and regularized Helmholtz--Hodge decompositions admit closed-form Fourier representations, enabling theory-aligned diagnostics. Across experiments E1--E6, GINO achieves low operator-approximation error, near machine-precision gauge equivariance, robustness to structured metric perturbations, strong cross-resolution generalization with small commutation error under restriction/prolongation, and structure-preserving performance on a regularized exact/coexact decomposition task. Ablations further link the smoothness of the learned spectral multiplier to stability under geometric perturbations. These results suggest that enforcing intrinsic structure and gauge equivariance yields operator surrogates that are more geometry-consistent and discretization-robust for elliptic PDEs on form-valued fields.

cs.AI↗