arXiv research

Improved perturbation reduces matrix condition number to O(n) with minimal storage.

New framework for higher-order singular-value derivatives of rectangular matrices.

The universal perturbative invariants of rational homology spheres can be extracted from the Chern-Simons partition function by combining perturbative and nonperturbative results. We spell out the general procedure to compute these invariants, and we work out in detail the case of Seifert spaces. By extending some prev…

Beyond existing multi-view clustering, this paper studies a more realistic clustering scenario, referred to as incomplete multi-view clustering, where a number of data instances are missing in certain views. To tackle this problem, we explore spectral perturbation theory. In this work, we show a strong link between per…

Complex Chern-Simons theory reveals peacock patterns in perturbative series.

Paper analyzes singular subspace estimation in noisy matrix models.

In this paper, we discuss the sensitivity of quantum PageRank. By using the finite dimensional perturbation theory, we estimate the change of the quantum PageRank under a small analytical perturbation on the Google matrix. In addition, we will show the way to estimate the lower bound of the convergence radius as well a…

Classical matrix perturbation results, such as Weyl's theorem for eigenvalues and the Davis-Kahan theorem for eigenvectors, are general purpose. These classical bounds are tight in the worst case, but in many settings sub-optimal in the typical case. In this paper, we present perturbation bounds which consider the natu…

Spectral methods simplify data analysis, improving accuracy and stability.

Paper addresses eigenvector perturbation in small eigen-gap scenarios.

New method estimates heterogeneous treatment effects with improved guarantees.

The paper examines how gradient descent stabilizes low-rank matrix factorization in noisy conditions.

New knot invariants derived using quantum cluster algebras.

The study examines collective behavior in banking sectors across mature and emerging markets.

Adding node feature kernels improves GCN robustness to graph perturbations.

This paper is concerned with the interplay between statistical asymmetry and spectral methods. Suppose we are interested in estimating a rank-1 and symmetric matrix $\mathbf{M}^{\star}\in \mathbb{R}^{n\times n}$, yet only a randomly perturbed version $\mathbf{M}$ is observed. The noise matrix $\mathbf{M}-\mathbf{M}^{\s…

Paper shows robustness of gradient descent in matrix sensing despite perturbations.

Paper develops a new kernel approximation framework.

Gradient descent with noise converges to a unique optimum in nonconvex matrix factorization.

We propose a general framework to study the stability of the subspace spanned by $P$ consecutive eigenvectors of a generic symmetric matrix ${\bf H}_0$, when a small perturbation is added. This problem is relevant in various contexts, including quantum dissipation (${\bf H}_0$ is then the Hamiltonian) and risk control …

Paper bounds subspace estimator error from noisy projections.

Study of regularized least squares in RKKS with indefinite kernels.

A new method for feature selection robust to noise and design variability.

Paper connects neural networks to Gaussian processes for understanding double-descent.

New technique stabilizes singular values in concatenated matrices.

The trivial flat connection's Chern-Simons theory is resurgent, revealing its structure.

Study analyzes perturbations in singular subspaces under random noise.

Gradient descent solves asymmetric low-rank matrix factorization efficiently.

The Davis-Kahan-Wedin $\sin Θ$ theorem describes how the singular subspaces of a matrix change when subjected to a small perturbation. This classic result is sharp in the worst case scenario. In this paper, we prove a stochastic version of the Davis-Kahan-Wedin $\sin Θ$ theorem when the perturbation is a Gaussian rando…

Paper proposes a method to generate adversarial perturbations for black-box attacks without accessing inner states.

We propose a new input perturbation mechanism for publishing a covariance matrix to achieve $(ε,0)$-differential privacy. Our mechanism uses a Wishart distribution to generate matrix noise. In particular, We apply this mechanism to principal component analysis. Our mechanism is able to keep the positive semi-definitene…

Unified PAC-Bayesian framework for deep learning generalization.

Identifies all perturbative vacua in bosonic string theory.

New bounds for private matrix approximation using Gaussian noise and Dyson Brownian Motion.

We develop an improved bound for the approximation error of the Nyström method under the assumption that there is a large eigengap in the spectrum of kernel matrix. This is based on the empirical observation that the eigengap has a significant impact on the approximation error of the Nyström method. Our approach is bas…

Spectral clustering is one of the most widely used techniques for extracting the underlying global structure of a data set. Compressed sensing and matrix completion have emerged as prevailing methods for efficiently recovering sparse and partially observed signals respectively. We combine the distance preserving measur…

Given a measurement graph $G= (V,E)$ and an unknown signal $r \in \mathbb{R}^n$, we investigate algorithms for recovering $r$ from pairwise measurements of the form $r_i - r_j$; $\{i,j\} \in E$. This problem arises in a variety of applications, such as ranking teams in sports data and time synchronization of distribute…

Develops a Gaussian model to compute the Alexander polynomial of knots.

Constructing an efficient parameterization of a large, noisy data set of points lying close to a smooth manifold in high dimension remains a fundamental problem. One approach consists in recovering a local parameterization using the local tangent plane. Principal component analysis (PCA) is often the tool of choice, as…

We consider aspects of Chern-Simons theory on L(p,q) lens spaces and its relation with matrix models and topological string theory on Calabi-Yau threefolds, searching for possible new large N dualities via geometric transition for non-SU(2) cyclic quotients of the conifold. To this aim we find, on one hand, some novel …

Robust tensor CP decomposition involves decomposing a tensor into low rank and sparse components. We propose a novel non-convex iterative algorithm with guaranteed recovery. It alternates between low-rank CP decomposition through gradient ascent (a variant of the tensor power method), and hard thresholding of the resid…

We developed a perturbation model for affine gravity theories.

New sigma models compute graviton scattering amplitudes from quaternionic geometry.

Perturbative string amplitudes are correctly derived from the string geometry theory, which is one of the candidates of a non-perturbative formulation of string theory. In order to derive non-perturbative effects rather easily, we formulate topological string geometry theory. We derive the perturbative partition functi…

We introduce a principled and theoretically sound spectral method for $k$-way clustering in signed graphs, where the affinity measure between nodes takes either positive or negative values. Our approach is motivated by social balance theory, where the task of clustering aims to decompose the network into disjoint group…

Study linear perturbations of Spin(7) metrics, finding only rank one nilpotent matrices.

Short proof shows how ridge regression works with random data.

The paper analyzes how quantization affects the Fisher Information Matrix's dominant eigenvalue.