Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2302.09658

iPCA and stability of star quivers

Cole Franks·Visu Makam
View original source ↗Read source document ↗Export metadata ↓

Abstract

Integrated principal components analysis, or iPCA, is an unsupervised learning technique for grouped vector data recently defined by Tang and Allen. Like PCA, iPCA computes new axes that best explain the variance of the data, but iPCA is designed to handle corrupting influences by the elements within each group on one another - e.g. data about students at a school grouped into classrooms. Tang and Allen showed empirically that regularized iPCA finds useful features for such grouped data in practice. However, it is not yet known when unregularized iPCA generically exists. For contrast, PCA (which is a special case of iPCA) typically exists whenever the number of data points exceeds the dimension. We study this question and find that the answer is significantly more complicated than it is for PCA. Despite this complexity, we find simple sufficient conditions for a very useful case - when the groups are no more than half as large as the dimension and the total number of data points exceeds the dimension, iPCA generically exists. We also fully characterize the existence of iPCA in case all the groups are the same size. When all groups are not the same size, however, we find that the group sizes for which iPCA generically exists are the integral points in a non-convex union of polyhedral cones. Nonetheless, we exhibit an algorithm to decide whether iPCA generically exists that is polynomial time in the node dimensions (based on the affirmative answer for the saturation conjecture by Knutson and Tao as well as a very simple randomized algorithm.At its core, our approach identifies connections between iPCA and stability notions for star quivers, thus bringing tools from invariant theory and quiver representations to the table.

Explore related subjects

math.STstat.TH

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Cole Franks, Visu Makam. 2023-07-01. iPCA and stability of star quivers. https://arxiv.org/abs/2302.09658

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Functional independent component analysis by choice of norm: a framework for near-perfect classification

We develop a theory for functional independent component analysis in an infinite-dimensional framework using Sobolev spaces that accommodate smoother functions. The notion of penalized kurtosis is introduced motivated by Silverman's method for smoothing principal components. This approach allows for a classical definition of independent components obtained via projection onto the eigenfunctions of a smoothed kurtosis operator mapping a whitened functional random variable. We discuss the theoretical properties of this operator in relation to a generalized Fisher discriminant function and the relationship it entails with the Feldman-Hájek dichotomy for Gaussian measures, both of which are critical to the principles of functional classification. The proposed estimators are a particularly competitive alternative in binary classification of functional data and can eventually achieve the so-called near-perfect classification, which is a genuine phenomenon of high-dimensional data. Our methods are illustrated through simulations, various real datasets, and used to model electroencephalographic biomarkers for the diagnosis of depressive disorder.

math.ST↗
arXiv2026

Trace-Class Results for MCMC Algorithms for Student-$t$ Regression Models

In this paper, we consider MCMC algorithms for Student-$t$ regression models. In three cases, we investigate the efficiency of Markov chains based on the algorithms in terms of whether trace-class results hold or not. First, we consider the case where the parameters follow a matrix-normal-inverse-Wishart distribution and show that the Markov operator associated with a standard data augmentation algorithm is trace-class. Second, we consider the case of an improper prior and univariate outcomes. In this case, the standard Markov operator is not trace-class but the Markov operator associated with a collapsed Gibbs algorithm is trace-class. Third, we consider the case of an improper prior and multivariate outcomes. We obtain a trace-class result for a parameter expanded data augmentation algorithm which is based on a univariate working parameter. Finally, we consider the problem of numerially estimating a convergence rate of the trace-class Markov operator in the second case.

math.ST↗
arXiv2026

The Manifold Hypothesis under Unknown Gaussian Noise:Conditional Certificates and Consistent Dimension Estimation

We study what noisy data can establish about the Manifold Hypothesis under explicit identification and regularity conditions. A population residual certificate combines independent-view localization, Gaussian concentration, membership uncertainty, and population transfer. Existing rectifiability criteria then yield a covered-scale consequence. For a local smooth manifold with positive Hölder density, the actual-ball covariance limit identifies the spectral crossing with geometric dimension. We prove almost-sure eventual recovery under repeated observations. Reusing accurate localization averages improves the sufficient point-sample condition from $Nr^{d+4}\gg\log N$ to $Nr^d\gg\log N$, with replication $kr^2\gg\log N$. A two-mass certificate controls incorrect geometric-dimension emissions under declared class bounds. For single observations with unknown Gaussian noise, affine-support or known coordinate-bound restrictions provide noise intervals and consistent Gaussian correlation-dimension estimators. Ahlfors regularity identifies this exponent with Hausdorff dimension and with the geometric dimension of a homogeneous smooth class. Exact Cantor calculations delineate the limits of integer spectral counts and adjacent-radius slopes. We credit established local PCA, rectifiability, concentration, binomial inference, and deconvolution results before specifying our constructions. Reproducible experiments distinguish point estimation, finite-scale coverage, and certificate emission.

math.ST↗