New methods rank players using covariates and comparisons, outperforming existing algorithms.
problem Ranking players based on incomplete and noisy pairwise comparisons.
method Three spectral ranking methods incorporating player covariates.
result Proposed methods outperform existing algorithms in simulations.
We describe a seriation algorithm for ranking a set of items given pairwise comparisons between these items. Intuitively, the algorithm assigns similar rankings to items that compare similarly with all others. It does so by constructing a similarity matrix from pairwise comparisons, using seriation methods to reorder t…
Partial convexification improves tractability of low-rank spectral optimization problems.
problem Minimizing linear objectives subject to matrix inequalities and low-rank constraints.
method Partial convexification of the domain set, deriving rank bounds, and developing a column generation algorithm.
result The partial convexification LSOP-R is equivalent to the original LSOP under certain conditions and yields high-quality solutions.
Spectral ranking methods are improved against semi-random graph sampling.
problem Improving spectral ranking methods in semi-random graph sampling.
method Investigating entry-wise error of spectral algorithms against a semi-random adversary.
result Asymptotic performance can be recovered by reweighting observed edges.
Deep networks learn clean structure before memorizing corrupted labels, leaving a spectral signature in gradient centered scatter.
problem Deep networks' transition from learning clean structure to memorizing corrupted labels under label noise.
method Analysis of the centered scatter of per-example last-layer gradients to identify Fisher Rank Inflation.
result Fisher Rank Inflation is a spectral signature of memorization under label noise, with effective rank expanding during memorization.
In this note, we present a new way to associate a spectral triple to the noncommutative C ∗ C^* C ∗ -algebra C ∗ ( Λ ) C^*(Λ) C ∗ ( Λ ) of a strongly connected finite higher-rank graph Λ Λ Λ . We generalize a spectral triple of Consani and Marcolli from Cuntz-Krieger algebras to higher-rank graph C ∗ C^* C ∗ -algebras C ∗ ( Λ ) C^*(Λ) C ∗ ( Λ ) , and we prove that these s…
The paper improves spectral ranking methods for diverse comparison graphs.
problem Estimating preference scores from multiway comparisons with heterogeneous sizes.
method Develops a two-step spectral method for estimating preference scores and their uncertainties.
result The two-step spectral method achieves the same asymptotic efficiency as the Maximum Likelihood Estimator (MLE).
This paper explores the preference-based top- K K K rank aggregation problem. Suppose that a collection of items is repeatedly compared in pairs, and one wishes to recover a consistent ordering that emphasizes the top- K K K ranked items, based on partially revealed preferences. We focus on the Bradley-Terry-Luce (BTL) model…
New spectral methods improve matrix estimation in RL with low-rank structure.
problem Estimating matrices with low-rank structure in reinforcement learning.
method Spectral-based matrix estimation approaches.
result Spectral methods efficiently recover singular subspaces and minimize entry-wise error.
The study characterizes wobbly rank-2 bundles on Riemann surfaces using spectral curves.
problem Characterizing wobbly rank-2 bundles on Riemann surfaces.
method Using spectral curves and direct images of line bundles, the study provides sufficient and necessary conditions for wobbly bundles.
result All rank-2 wobbly bundles can be characterized as twists of direct images of line bundles.
Paper improves robust spectral clustering for noisy data.
problem Noisy data and heavy-tailed entries hinder traditional clustering methods.
method Robust spectral clustering with rank statistics for latent structure recovery.
result Provable recovery of latent block structure in large data matrices.
Gradient descent with small random init mimics spectral methods for low-rank matrix recovery.
problem Reconstructing a low-rank matrix from few measurements.
method Gradient descent with small random initialization followed by a few iterations.
result Gradient descent from small random init converges to a well-generalizing solution.
Improved rank aggregation via spectral method reduces sample complexity.
problem Ranking items from pairwise comparisons with corrupted data.
method Spectral ranking algorithms based on unnormalized and normalized data matrices.
result Sharper ℓ ∞ \ell_{\infty} ℓ ∞ -norm perturbation bound and error bound on maximum displacement for each item. We introduce a new parameterization method for deep learning layers using spectral tensor train decomposition.
problem Efficiency and stability in deep learning models with weight matrix compression.
method Spectral Tensor Train Parameterization (STTP) of weight matrices.
result Improved compression and training stability in neural networks.
Spectral gradient methods outperform Euclidean in certain deep learning scenarios.
problem When do spectral gradient updates outperform Euclidean in deep learning?
method Layerwise condition comparing squared nuclear-to-Frobenius ratio to stable rank of activations.
result Spectral updates can be more effective than Euclidean in deep networks and transformers.
Study finite group actions on 4-manifolds, finding rank bounds.
problem Understanding finite group actions on 4-manifolds.
method Investigate Borel spectral sequence for G-equivariant cohomology.
result Establish new bounds on the rank of G for homologically trivial actions.
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.
Study Higgs bundles on curves with punctures, extending spectral correspondence.
problem Classify Higgs bundles on punctured curves with logarithmic structures.
method Logarithmic Hecke compactification, spectral conditions, and sheaf classification.
result Logarithmic spectral correspondence extended to punctured curves.
Solves the Wiegold problem by showing free products of left-orderable groups have normal rank > 1.
problem Wiegold problem about groups of normal rank > 1
method Topological argument and intricate construction of left-orders
result Free products of nontrivial left-orderable groups have normal rank > 1
Study spectral flow on a warped cylinder with special boundary conditions.
problem Analyzing spectral flow on a warped cylinder with specific boundary conditions.
method Complexifying the twisting bundle, diagonalizing the orthogonal twist, and regrouping conjugate and reflection-paired blocks.
result Explicit formula for R O ( O ( 2 ) ) RO(O(2)) R O ( O ( 2 )) -valued spectral flow, refining ordinary spectral flow. Paper improves MVSC using tensor low-rank modeling.
problem Improving multi-view spectral clustering.
method Structured tensor low-rank norm for MVSC optimization.
result Proposed method outperforms state-of-the-art methods.
New algorithm learns low-rank matrices with linear number of samples.
problem Learning low-rank matrices efficiently in latent-variable applications.
method Proposed algorithm that uses linear number of samples in high dimension.
result Learning k i m e s k k imes k k im es k , rank- r r r , matrices requires $Ω( rac{kr}{ε^2})$ samples. Study critical exponents for L^p-cohomology of higher rank Lie groups and manifolds.
problem Investigate critical exponents for vanishing L^p-cohomology in higher rank Lie groups and manifolds.
method Examine SL 3 _3 3 (R) and 5-dimensional solvable Lie groups, use spectral sequence arguments. result Discover a continuum of quasi-isometry classes of rank 2 solvable Lie groups.
A well-known conjecture of Rasmussen states that for any knot K K K in S 3 S^{3} S 3 , the rank of the reduced Khovanov homology of K K K is greater than or equal to the rank of the reduced knot Floer homology of K K K . This rank inequality is supposed to arise as the result of a spectral sequence from Khovanov homology to knot Flo…
New method improves matrix completion accuracy, especially in noisy data.
problem Noisy matrix completion in recommendation systems and signal processing.
method Residual Spectral Matching criterion and pseudo-gradient algorithms.
result Improved numerical performance in noisy data environments.
Unified view of spectral networks linking geometry and gauge theory.
problem Understanding BPS states in gauge theories.
method Unified geometric and physical approaches, focusing on spectral networks.
result Spectral networks provide a framework for determining BPS spectra.
Constructs coordinate systems from spectral curve sheaves.
problem Creating coordinate systems from spectral curve sheaves.
method Finite-gap integration methods for orthogonal curvilinear coordinates.
result Constructs coordinate systems over reducible spectral curves.
New method for initializing low-rank neural networks improves performance.
problem Training low-rank neural networks efficiently and accurately.
method Inspired by function approximation, proposes a novel low-rank initialization framework.
result Demonstrates significant gap between spectral and low-rank initialization approaches.
Let E k F ( D ) E_{k}^{F}(D) E k F ( D ) be the spectral sequence induced by the oriented cube of resolutions on knot Floer homology. We prove that E 2 F ( D ) E_{2}^{F}(D) E 2 F ( D ) is a triply graded link invariant whose graded Euler characteristic is the HOMFLY-PT polynomial and that the higher pages are link invariants. By construction, the spectral sequen…
Optimizes ranking of top-k players from partial comparison data.
problem Identifying the top-k players from incomplete pairwise comparisons.
method Maximum Likelihood Estimator (MLE) and Spectral Method.
result MLE achieves optimal partial and exact recovery, while Spectral Method is sub-optimal.
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.
Develops methods to estimate high rank tensors from noisy data.
problem Estimating high rank tensors from noisy observations.
method Generative latent variable tensor model, polynomial-time spectral algorithm.
result Achieves computationally optimal rate for signal tensor estimation.
Study on harmonic metrics for rank 3 Higgs bundles in Hitchin section.
problem Finding compatible harmonic metrics for rank 3 Higgs bundles in the Hitchin section.
method Defined a symmetric pairing and studied spectral curves as 2-sheeted branched coverings.
result Gave a condition for Higgs bundles on C \mathbb{C} C or C ∗ \mathbb{C}^* C ∗ to have compatible harmonic metrics. Consider the problem of estimating a low-rank matrix when its entries are perturbed by Gaussian noise. If the empirical distribution of the entries of the spikes is known, optimal estimators that exploit this knowledge can substantially outperform simple spectral approaches. Recent work characterizes the asymptotic acc…
The spectral k k k -support norm enjoys good estimation properties in low rank matrix learning problems, empirically outperforming the trace norm. Its unit ball is the convex hull of rank k k k matrices with unit Frobenius norm. In this paper we generalize the norm to the spectral ( k , p ) (k,p) ( k , p ) -support norm, whose additional para…
We explore the top- K K K rank aggregation problem. Suppose a collection of items is compared in pairs repeatedly, and we aim to recover a consistent ordering that focuses on the top- K K K ranked items based on partially revealed preference information. We investigate the Bradley-Terry-Luce model in which one ranks items ac…
We accelerate the power method for strong low-rank approximation using fast sketching.
problem Efficiency bottleneck in power method for large target ranks.
method Developed an algorithmic and theoretical framework for accelerating the power method using fast sketching.
result Simple and provably efficient methods for singular value decomposition, low-rank factorization, and Nyström approximation.
The paper improves tensor completion bounds using spectral gap.
problem Theoretical limitations in tensor completion, especially for deterministic sampling.
method Bounding the generalization error of tensor completion methods using spectral gap.
result Improved bounds on tensor completion error, reducing rank dependence.
New method ranks sectors and countries using local and aggregate I-O data.
problem Ranking sectors and countries in global value chains using incomplete I-O tables.
method Rank- 1 1 1 approximation to I-O tables using local and aggregate information. result Consistently good performance in reconstructing rankings of upstreamness and downstreamness.
For a 2-periodic link L ~ \tilde L L ~ in the thickened annulus and its quotient link L L L , we exhibit a spectral sequence with E 1 ≅ A K h ( L ~ ) ⊗ F 2 F 2 [ θ , θ − 1 ] ⇉ E ∞ ≅ A K h ( L ) ⊗ F 2 F 2 [ θ , θ − 1 ] . E^1 \cong AKh(\tilde L) \otimes_{\mathbb{F}_2} \mathbb{F}_2[θ, θ^{-1}] \rightrightarrows E^\infty \cong AKh(L) \otimes_{\mathbb{F}_2} \mathbb{F}_2[θ, θ^{-1}]. E 1 ≅ A K h ( L ~ ) ⊗ F 2 F 2 [ θ , θ − 1 ] ⇉ E ∞ ≅ A K h ( L ) ⊗ F 2 F 2 [ θ , θ − 1 ] . This spectral sequence splits along qu…
This paper proposes a new Nystrom-based clustering algorithm for large-scale data.
problem Spectral clustering's high computational complexity for large-scale data.
method Centroid Minimum Sum of Squared Similarities (CMS3) sampling procedure with eigen spectrum shape heuristic.
result Competitive low-rank approximations in test datasets compared to state-of-the-art methods.
The paper classifies links with low rank knot Floer and Khovanov homologies.
problem Detecting and classifying links with low rank knot Floer and Khovanov homologies.
method Generalized link Floer homology, used to obtain rank bounds and classify links.
result Knot Floer homology detects T ( 2 , 8 ) T(2,8) T ( 2 , 8 ) and T ( 2 , 10 ) T(2,10) T ( 2 , 10 ) . We show that the spectral norm of a random n 1 × n 2 × ⋯ × n K n_1\times n_2\times \cdots \times n_K n 1 × n 2 × ⋯ × n K tensor (or higher-order array) scales as O ( ( ∑ k = 1 K n k ) log ( K ) ) O\left(\sqrt{(\sum_{k=1}^{K}n_k)\log(K)}\right) O ( ( ∑ k = 1 K n k ) log ( K ) ) 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…
This paper improves spectral clustering for large datasets using the Nystrom method.
problem Spectral clustering's scalability issues with large datasets.
method A principled spectral clustering algorithm exploiting Nystrom approximation's spectral properties.
result Improved spectral clustering efficiency and accuracy compared to existing methods.
Batch normalization prevents rank collapse in deep networks, improving training stability.
problem Rank collapse in randomly initialized deep networks with increasing depth.
method Investigates spectral instabilities in random matrices and uses batch normalization to avoid rank collapse.
result Batch normalization prevents rank collapse in both linear and ReLU networks, improving training stability.
This study analyzes why attention layers in neural networks can cause signal loss and proposes a solution.
problem Pathological behavior of attention layers in neural networks, leading to signal loss.
method Spectral analysis using Random Matrix Theory to identify and mitigate rank collapse in width.
result A novel solution to mitigate rank collapse in width by removing outlier eigenvalues.
Given a graphical model (GM), computing its partition function is the most essential inference task, but it is computationally intractable in general. To address the issue, iterative approximation algorithms exploring certain local structure/consistency of GM have been investigated as popular choices in practice. Howev…
Paper introduces a novel framework for recognizing dynamic ranking structures in preference-based data.
problem Complex and noisy preference-based data often hide underlying homogeneous structures.
method Developed an approach to identify dynamic ranking groups using temporal penalties and spectral estimation. Introduced an objective function for detecting structural changes.
result Consistent recognition of ranking groups and structural changes in preference-based data.