arXiv research

Spectral algorithms are graph partitioning algorithms that partition a node set of a graph into groups by using a spectral embedding map. Clustering techniques based on the algorithms are referred to as spectral clustering and are widely used in data analysis. To gain a better understanding of why spectral clustering i…

We obtain a Bonnet-Myers theorem under a spectral condition: a closed Riemannian manifold $(M^n,g)$ for which the lowest eigenvalue of the Ricci tensor $ρ$ is such that the Schrödinger operator $(n-2)Δ+ ρ$ is positive has finite fundamental group. As a continuation of our earlier results, we obtain isoperimetric inequa…

A new method for nonstationary Gaussian processes using Fourier features.

Unified framework for analyzing graph neural operators converging to graph limits.

New method extrapolates spectral densities from smaller models to larger ones.

We give a polynomial-time algorithm for learning latent-state linear dynamical systems without system identification, and without assumptions on the spectral radius of the system's transition matrix. The algorithm extends the recently introduced technique of spectral filtering, previously applied only to systems with a…

The paper proves conditions for Kähler and Riemannian manifolds to be simply connected.

Spectral Independence Criterion helps infer cause-effect relationships in time series.

Define mass invariant for asymptotically hyperbolic 3-manifolds with toroidal infinity.

The paper analyzes neural networks for solving high-dimensional Schrödinger eigenvalue problems.

Improved model for non-smooth signals with complex spectra.

Study improves understanding of Ricci curvature in manifolds.

We present a fairly general construction of unbounded representatives for the interior Kasparov product. As a main tool we develop a theory of C^1-connections on operator * modules; we do not require any smoothness assumptions; our sigma-unitality assumptions are minimal. Furthermore, we use work of Kucerovsky and our …

New methods rank players using covariates and comparisons, outperforming existing algorithms.

Under appropriate spectral assumptions we prove two existence results for positive solutions of Lichnerowicz-type equations on complete manifolds. We also give a priori bounds and a comparison result that immediately yields uniqueness for certain classes of solutions. No curvature assumptions are involved in our analys…

Free lunch from noise reveals linear spectral features for RL.

Sharp spectral theorem splits certain non-compact manifolds.

Under a spectral assumption on the Laplacian of a Poincaré--Einstein manifold, we establish an energy inequality relating the energy of a fractional GJMS operator of order $2γ\in(0,2)$ or $2γ\in(2,4)$ and the energy of the weighted conformal Laplacian or weighted Paneitz operator, respectively. This spectral assumption…

New bounds for KRR condition number reveal overfitting phenomena.

A large number of algorithms in machine learning, from principal component analysis (PCA), and its non-linear (kernel) extensions, to more recent spectral embedding and support estimation methods, rely on estimating a linear subspace from samples. In this paper we introduce a general formulation of this problem and der…

New spectral clustering method for graphs with uneven node degrees.

We consider a distributed learning approach in supervised learning for a large class of spectral regularization methods in an RKHS framework. The data set of size n is partitioned into $m=O(n^α)$ disjoint subsets. On each subset, some spectral regularization method (belonging to a large class, including in particular K…

In this paper we study some new von Neumann spectral invariants associated to the Laplacian acting on L^2 differential forms on the universal cover of a closed manifold. These invariants coincide with the Novikov-Shubin invariants whenever there is no spectral gap in the spectrum of the Laplacian, and are homotopy inva…

We establish a splitting formula for the spectral flow of the odd signature operator on a closed 3-manifold M coupled to a path of SU(2) connections, provided M = S cup X, where S is the solid torus. It describes the spectral flow on M in terms of the spectral flow on S, the spectral flow on X (with certain Atiyah-Pato…

Study shows consistency of shallow GCNNs on sampled point clouds under manifold assumption.

Spectral methods predict long-term signals from linear and nonlinear systems.

Researchers create spectral triples for twisted crossed products using Kasparov's external product.

Improved LDA method for better classification and dimensionality reduction.

FASC clusters data with latent factors, improving on naive methods.

LASE improves local network structure visualization by targeting locally low-dimensional regions.

Study Bergman and spectral kernels for non-compact complex manifolds.

We show that the spectral norm of a random $n_1\times n_2\times \cdots \times n_K$ tensor (or higher-order array) scales as $O\left(\sqrt{(\sum_{k=1}^{K}n_k)\log(K)}\right)$ under some sub-Gaussian assumption on the entries. The proof is based on a covering number argument. Since the spectral norm is dual to the tensor…

We analyze DMs using spectral methods to design effective noise schedules.

Spectral clustering is a fast and popular algorithm for finding clusters in networks. Recently, Chaudhuri et al. (2012) and Amini et al.(2012) proposed inspired variations on the algorithm that artificially inflate the node degrees for improved statistical performance. The current paper extends the previous statistical…

Reconstruction based subspace clustering methods compute a self reconstruction matrix over the samples and use it for spectral clustering to obtain the final clustering result. Their success largely relies on the assumption that the underlying subspaces are independent, which, however, does not always hold in the appli…

Wave operators and spectral stability for Dirac operators under Ricci flow.

SMART transfers knowledge across related studies for multi-task learning.

The dictionary-aided sparse regression (SR) approach has recently emerged as a promising alternative to hyperspectral unmixing (HU) in remote sensing. By using an available spectral library as a dictionary, the SR approach identifies the underlying materials in a given hyperspectral image by selecting a small subset of…

New method optimizes portfolios for non-stationary markets.

Spectral density matrix estimation of multivariate time series is a classical problem in time series and signal processing. In modern neuroscience, spectral density based metrics are commonly used for analyzing functional connectivity among brain regions. In this paper, we develop a non-asymptotic theory for regularize…

A framework uses free probability to analyze Transformer models.

Paper shows stability of metric reconstruction for orbifolds from spectral data.

Deep learning speeds spectral density estimation for large 2D/3D grids.

Most successful machine intelligence systems rely on gradient-based learning, which is made possible by backpropagation. Some systems are designed to aid us in interpreting data when explicit goals cannot be provided. These unsupervised systems are commonly trained by backpropagating through a likelihood function. We i…

This research introduces dynamic portfolio cuts using a spectral approach for graph-theoretic diversification.

Estimates linear model from noisy covariates and instruments using spectral regularization.

Paper finds a counter-example invalidating a spectral asymptotic algorithm.

Method constrains spectral gaps of hyperbolic spin surfaces using identities and semidefinite programming.