Optimizes risk measures given known marginal distributions of two unknown factors.
problem Determining an upper bound for spectral risk measures with unknown joint distribution.
method Introduces Maximum Spectral Measure (MSP) as a worst-case risk measure, formulated as an optimization problem with a more general objective function.
result Characterizes the continuity properties of the optimal value function and optimal solution set with respect to marginal distributions.
FASC clusters data with latent factors, improving on naive methods.
problem Clustering high-dimensional data with correlated variables.
method Factor Adjusted Spectral Clustering (FASC) algorithm.
result FASC achieves an exponentially low mislabeling rate under general assumptions.
SG-NTF completes HDI tensors with spectral mapping and spatio-temporal gating.
problem High-dimensional and incomplete tensor completion.
method Spectra-Guided Neural Tucker Factorization (SG-NTF) with Spatio-Temporal Co-Gating (STCG).
result Maintains competitive completion accuracy with parameter efficiency.
Estimates linear model from noisy covariates and instruments using spectral regularization.
problem Estimating a linear model from many noisy covariates and instruments.
method Two-stage least squares with spectral regularization of canonical correlations.
result Upper and lower bounds on estimation error, proving optimality of the method with noisy data.
Deep learning method clusters multi-view data matrices.
problem Clustering heterogeneous relational data matrices.
method Deep collective matrix tri-factorization (DCMTF).
result Discover latent clusters across input matrices and their associations.
The paper examines how gradient descent stabilizes low-rank matrix factorization in noisy conditions.
problem Stability of low-rank implicit regularization in perturbed deep matrix factorization.
method Derives spectral conditions for gradient descent to exhibit a low-rank phase in noiseless settings and analyzes perturbed dynamics.
result Gradient descent converges to a low-rank solution under perturbation, with explicit dependence on perturbation size.
Spectral clustering identifies clusters of multivariate extremes.
problem Analyzing the dependence structure of multivariate extremes.
method Spectral clustering based on a random k-nearest neighbor graph. result Spectral clustering can consistently identify clusters of multivariate extremes under certain conditions.
This work connects LLE, factor analysis, and probabilistic PCA through a stochastic perspective.
problem Exploring the theoretical connection between LLE, factor analysis, and probabilistic PCA.
method Solving the stochastic linear reconstruction of LLE using expectation maximization.
result LLE, factor analysis, and probabilistic PCA are shown to be connected through a stochastic perspective.
The Ray-Singer isospectral theorem (1971) is applied to a general spectral function for Laplacians of twisted p-forms (say) on homogeneous Clifford-Klein factors of the three-sphere. The inducing formulae necessary to express any spectral quantity for any twisting in terms of those for cyclic subgroups of the tetrahedr…
Deep neural networks decompose SDF into linear and nonlinear components.
problem Constructing accurate stochastic discount factors (SDFs) for pricing.
method Additive decomposition of a deep neural network trained to construct SDFs.
result The PTK representation delivers significant performance gains in equity data.
Method estimates shared and study-specific factors for multi-study data.
problem Covariance estimation for multi-study data with shared and study-specific components.
method Spectral decomposition for latent factors, surrogate Bayesian regressions for loadings and variances.
result Strong frequentist guarantees and superior performance in simulations and real data.
Paper finds exact Hessian sharpness in deep matrix factorization.
problem Understanding the geometry of loss landscapes in deep matrix factorization.
method Presented the first exact expression for Hessian maximum eigenvalue.
result Spectral-norm balance is a sufficient condition for flatness in deep matrix factorization.
Irregular features disrupt the desired classification. In this paper, we consider aggressively modifying scales of features in the original space according to the label information to form well-separated clusters in low-dimensional space. The proposed method exploits spectral clustering to derive scaling factors that a…
New method efficiently learns positive-definite curvature for neural nets.
problem Efficiently learn positive-definite curvature for neural net training.
method Spectral-factorized positive-definite curvature learning approach.
result Efficiently applies arbitrary matrix roots and generic curvature learning.
Characterizes bi-Perron numbers with specific Galois conjugates.
problem Understanding bi-Perron numbers with real or unimodular conjugates.
method Characterization through power properties and spectral radii of transformations.
result Bi-Perron numbers with real or unimodular conjugates admit a power as stretch factors or spectral radii.
New method detects global factors near BBP phase transition in high-dimensional data.
problem Detecting the number of global factors in noisy high-dimensional correlation matrices.
method Iterative Global Factor (IGF) algorithm combining adaptive edge recalibration and PR delocalization filter.
result IGF algorithm successfully detects global factors near BBP transition, improving over eigenvalue-only methods.
Study examines metrics and functionals for Hodge-Dirac operator on manifolds.
problem Examining metrics and functionals for Hodge-Dirac operator on manifolds.
method Analyzing the metric and Einstein bilinear functionals of differential forms for Hodge-Dirac operator d+δ. result The functionals reproduce those for the canonical Dirac operator on a spin manifold up to a numerical factor.
Let φ(G) be the minimum conductance of an undirected graph G, and let 0=λ_1 <= λ_2 <=... <= λ_n <= 2 be the eigenvalues of the normalized Laplacian matrix of G. We prove that for any graph G and any k >= 2, φ(G) = O(k) λ_2 / \sqrt{λ_k}, and this performance guarantee is achieved by the spectral partitioning algorithm. …
Spectral method for joint community detection and group synchronization.
problem Jointly detecting communities and synchronizing orthogonal groups in graphs.
method Spectral decomposition followed by CPQR factorization.
result Near-optimal guarantees for exact and stable recovery of cluster memberships and orthogonal transforms.
Improved performance of factorized neural layers through spectral initialization and Frobenius decay.
problem Improving the performance of factorized neural layers in various deep learning contexts.
method Spectral initialization and Frobenius decay for initialization and regularization.
result Spectral initialization and Frobenius decay lead to improved performance across multiple deep learning settings.
In deep neural networks, the spectral norm of the Jacobian of a layer bounds the factor by which the norm of a signal changes during forward/backward propagation. Spectral norm regularizations have been shown to improve generalization, robustness and optimization of deep learning methods. Existing methods to compute th…
We present a method based on the orthogonal symmetric non-negative matrix tri-factorization of the normalized Laplacian matrix for community detection in complex networks. While the exact factorization of a given order may not exist and is NP hard to compute, we obtain an approximate factorization by solving an optimiz…
EigenBayes: A fast, adaptive Bayesian shrinkage approach for high-dimensional matrix factorization
problem Choosing the latent dimension k in factor models method Adaptive spectral shrinkage and empirical Bayes calibration
result Adapts to signal-to-noise ratio and shrinks superfluous components
We show that, when considering the scaling factor as an affine variable, the coefficients of the asymptotic expansion of the spectral action on a (Euclidean) Robertson-Walker spacetime are periods of mixed Tate motives, involving relative motives of complements of unions of hyperplanes and quadric hypersurfaces and div…
We consider the problem of estimating a consensus community structure by combining information from multiple layers of a multi-layer network using methods based on the spectral clustering or a low-rank matrix factorization. As a general theme, these "intermediate fusion" methods involve obtaining a low column rank matr…
New method estimates covariance in multi-view data with better accuracy and uncertainty.
problem Estimating covariance in multi-view data with shared and view-specific latent factors.
method Spectral decompositions and conditional conjugate priors for factor loadings and residual variances.
result Proves favorable asymptotic properties and excellent performance in simulations and real data.
New bounds adaptively control spectral complexity of trained Transformers.
problem Understanding why Transformers generalize well in machine learning.
method Spectrum-adaptive post hoc generalization bounds for multi-layer Transformers.
result Bounds adaptively trade off spectral complexity against dimension and depth factors.
New spectral clustering method handles discrete covariates for better community detection.
problem Community detection in networks with discrete covariates.
method Spectral algorithm that separates latent network structure from observed covariates.
result Achieves perfect clustering with high probability in large, sparse networks.
Efficiently factorize tensors in streaming data with coreset selection.
problem Efficiently factorize tensors in streaming data.
method Online filtering and kernelization techniques to select a coreset of vectors.
result CP decomposition of coreset approximates full data tensor decomposition.
The study of spectral-tightness in Riemannian manifolds and its topological implications.
problem Understanding the spectral properties of Riemannian manifolds and their coverings.
method Analyzing the fundamental group and the Euclidean local de Rham factor to characterize spectral-tightness.
result Spectral-tightness is a topological property of the fundamental group, and it can be characterized by the dimension of the Euclidean local de Rham factor.
Advances neural tri-factorization for clustering and discordance analysis of multi-typed data.
problem Challenges in analyzing heterogeneous, multimodal relational data.
method Deep collective matrix tri-factorization for spectral clustering and cluster association learning.
result Demonstrates efficacy over previous non-neural approaches in clustering and discordance analysis.
We study the problem of large-scale network embedding, which aims to learn latent representations for network mining applications. Previous research shows that 1) popular network embedding benchmarks, such as DeepWalk, are in essence implicitly factorizing a matrix with a closed form, and 2)the explicit factorization o…
Tensor factorization arises in many machine learning applications, such knowledge base modeling and parameter estimation in latent variable models. However, numerical methods for tensor factorization have not reached the level of maturity of matrix factorization methods. In this paper, we propose a new method for CP te…
SON-NMF estimates nonnegative rank on-the-fly for NMF.
problem Estimating the nonnegative rank of data in NMF.
method Sum-of-norms (SON) regularization to reduce rank, combined with a first-order BCD algorithm.
result SON-NMF can automatically estimate the rank from data without prior knowledge.
This paper presents a margin-based multiclass generalization bound for neural networks that scales with their margin-normalized "spectral complexity": their Lipschitz constant, meaning the product of the spectral norms of the weight matrices, times a certain correction factor. This bound is empirically investigated for…
We consider the problem of clustering a set of high-dimensional data points into sets of low-dimensional linear subspaces. The number of subspaces, their dimensions, and their orientations are unknown. We propose a simple and low-complexity clustering algorithm based on thresholding the correlations between the data po…
A recent theoretical analysis shows the equivalence between non-negative matrix factorization (NMF) and spectral clustering based approach to subspace clustering. As NMF and many of its variants are essentially linear, we introduce a nonlinear NMF with explicit orthogonality and derive general kernel-based orthogonal m…
We study minimal annuli in S2×R of finite type by relating them to harmonic maps C→S2 of finite type. We rephrase an iteration by Pinkall-Sterling in terms of polynomial Killing fields. We discuss spectral curves, spectral data and the geometry of the isospectral set…
Seidel and Smith introduced the graded fixed-point symplectic Khovanov cohomology group Kh_{symp,inv}(K) for a knot K inside S^{3}, as well as a spectral sequence converging to the Heegaard Floer homology-hat group for the connected sum of the double branched cover with a copy of S^{2}xS^{1}. The E^{1}-page of this spe…
Muon optimizer simplifies matrix optimization with spectral orthogonalization.
problem Matrix optimization challenges, especially with large condition numbers.
method Simplified Muon optimizer using spectral orthogonalization of gradients.
result Simplified Muon converges linearly with independent scalar sequences, outperforming gradient descent and Adam.
Spectral portfolio theory links neural networks to wealth dynamics via SGD weight matrices.
problem Understanding wealth dynamics from neural network training.
method Direct identification of weight matrices as portfolio allocation matrices, linking SGD forces to portfolio dynamics.
result Spectral properties of SGD weight matrices transition between additive and multiplicative regimes, influencing wealth dynamics.
New method for analyzing complex data spaces.
problem Dimensionality reduction and learning data representations for continuous spaces.
method Manifold factorization based on spectral graph methods.
result Recovering factors yields meaningful lower-dimensional representations.
This paper proves a generalization bound for complex-valued neural networks scaling with spectral complexity.
problem Ensuring the performance of complex-valued neural networks on unseen data.
method Theoretical derivation using Maurey Sparsification Lemma and Dudley Entropy Integral, empirical validation on various datasets.
result The spectral complexity of weight matrices is a significant factor in the generalization ability of complex-valued neural networks.
KAN-PCA improves asset return analysis by capturing more variance than classical PCA during market crises.
problem Inefficient classical PCA during market crises when correlations between assets change dramatically.
method KAN-PCA uses KAN (Kolmogorov-Arnold Networks) with B-spline functions to learn nonlinear projections.
result KAN-PCA achieves a higher reconstruction R^2 (66.57%) compared to classical PCA (62.99%) on 20 S&P 500 stocks.
We study the spectral gap of the Erdős--Rényi random graph through the connectivity threshold. In particular, we show that for any fixed δ>0 if p≥n(1/2+δ)logn, then the normalized graph Laplacian of an Erdős--Rényi graph has all of its nonzero eigenvalues tightly concentrated around 1. We est…
Gradient descent solves asymmetric low-rank matrix sensing without balancing.
problem Recovering asymmetric low-rank matrices from linear measurements.
method Gradient descent with spectral initialization, avoiding balancing term.
result Gradient descent converges linearly without balancing, factors stay balanced.
Spectral feature learning improves IV regression for causal effect estimation.
problem Estimating causal effects in the presence of hidden confounders.
method Two-stage least squares estimator based on spectral features.
result Performance of the method depends on strong spectral alignment and slow eigenvalue decay.
Robo-advisors estimate clients' risk aversion using interactive questionnaires.
problem Estimating risk aversion of non-expert clients using adaptive questionnaires.
method Model risk aversion with cost functions and spectral risk measures. Use inverse reinforcement learning to design questions maximizing distinguishing power.
result Designing questions by maximizing distinguishing power achieves satisfactory accuracy in learning risk aversion with fewer than 50 questions.