arXiv research

A new dynamical formulation of log-PCA captures local principal modes of geodesic variations.

Study nonholonomic systems with collisions using variational principles.

We consider a contracting problem in which a principal hires an agent to manage a risky project. When the agent chooses volatility components of the output process and the principal observes the output continuously, the principal can compute the quadratic variation of the output, but not the individual components. This…

We introduce a new convex formulation for stable principal component pursuit (SPCP) to decompose noisy signals into low-rank and sparse representations. For numerical solutions of our SPCP formulation, we first develop a convex variational framework and then accelerate it with quasi-Newton methods. We show, via synthet…

We consider the second variational derivative of a given gauge-natural invariant Lagrangian taken with respect to (prolongations of) vertical parts of gauge-natural lifts of infinitesimal principal automorphisms. By requiring such a second variational derivative to vanish, {\em via} the Second Noether Theorem we find t…

MCPCA analyzes shared factors across multiple data contexts.

The paper presents new formulations of gauge and gravity theories using dynamical principal bundles.

When a gauge-natural invariant variational principle is assigned, to determine {\em canonical} covariant conservation laws, the vertical part of gauge-natural lifts of infinitesimal principal automorphisms -- defining infinitesimal variations of sections of gauge-natural bundles -- must satisfy generalized Jacobi equat…

Paper finds principal eigenvalue for infinity Laplacian in metric spaces.

A reductive structure is associated here with Lagrangian canonically defined conserved quantities on gauge-natural bundles. Parametrized transformations defined by the gauge-natural lift of infinitesimal principal automorphisms induce a variational sequence such that the generalized Jacobi morphism is naturally self-ad…

We compute first variation formulas for the complex components of the Bakry-Emery-Ricci endomorphism along Kähler structures. Our formulas show that the principal parts of the variations are quite standard complex differential operators with particular symmetry properties on the complex decomposition of the variation o…

Survey of factor analysis, PCA, variational inference, and VAE.

In this document we are going to derive the equations needed to implement a Variational Bayes i-vector extractor. This can be used to extract longer i-vectors reducing the risk of overfittig or to adapt an i-vector extractor from a database to another with scarce development data. This work is based on Patrick Kenny's …

Develops statistical framework for analyzing functional data extremes.

GP-PCA reduces infinite-dimensional GP posteriors to a finite space for meta-learning.

A new method improves target selection for manipulating complex systems like the brain.

We employ unsupervised machine learning techniques to learn latent parameters which best describe states of the two-dimensional Ising model and the three-dimensional XY model. These methods range from principal component analysis to artificial neural network based variational autoencoders. The states are sampled using …

In this work, we develop a novel principal component analysis (PCA) for semimartingales by introducing a suitable spectral analysis for the quadratic variation operator. Motivated by high-dimensional complex systems typically found in interest rate markets, we investigate correlation in high-dimensional high-frequency …

Bayesian SPCA method tackles orthogonality constraint with spike and slab prior.

Study explores K-means clustering of variables and its relation to PCA.

We study the existence and properties of metrics maximising the first Laplace eigenvalue among conformal metrics of unit volume on Riemannian surfaces. We describe a general approach to this problem and its higher eigenvalue versions via the direct method of calculus of variations. The principal results include the gen…

We classify all rotational surfaces in Euclidean space whose principal curvatures $κ_1$ and $κ_2$ satisfy the linear relation $κ_1=aκ_2+b$, where $a$ and $b$ are two constants. We give a variational characterization of these surfaces in terms of its generating curve. As a consequence of our classification, we find clos…

Regularized MFPCA smooths multivariate functional data for clearer patterns.

Rotates MFVI for better Gaussian approximations.

Study of deformations of Virasoro symmetries using variational bihamiltonian cohomology.

New method optimizes PCA for better prediction and variance.

This paper develops GPCA for probability distributions using Otto-Wasserstein geometry.

Unified method to compute Laplace spectra on homogeneous principal bundles.

Polynomial chaos expansions on Grassmannian submanifolds for high-dimensional stochastic systems.

Invariants found for tau-symmetric bihamiltonian systems.

Improves variational inference for sparse models using mixtures of exponential families.

Paper develops efficient AltMin algorithm for SRPCP robust matrix recovery.

This work improves disentanglement in latent space models without sacrificing generation quality.

The paper studies constant mean curvature hypersurfaces in Finsler manifolds.

We study sparse principal components analysis in high dimensions, where $p$ (the number of variables) can be much larger than $n$ (the number of observations), and analyze the problem of estimating the subspace spanned by the principal eigenvectors of the population covariance matrix. We introduce two complementary not…

We derive both {\em local} and {\em global} generalized {\em Bianchi identities} for classical Lagrangian field theories on gauge-natural bundles. We show that globally defined generalized Bianchi identities can be found without the {\em a priori} introduction of a connection. The proof is based on a {\em global} decom…

Extends Yang-Mills theory to non-integrable Lie algebroids.

In the recent years, Riemannian shape analysis of curves and surfaces has found several applications in medical image analysis. In this paper we present a numerical discretization of second order Sobolev metrics on the space of regular curves in Euclidean space. This class of metrics has several desirable mathematical …

The literature provides strong evidence that stock prices can be predicted from past price data. Principal component analysis (PCA) is a widely used mathematical technique for dimensionality reduction and analysis of data by identifying a small number of principal components to explain the variation found in a data set…

New method for hyperparameter tuning in sparse matrix factorization.

Generalizes Carathéodory form for higher-order field theories.

We focus on the robust principal component analysis (RPCA) problem, and review a range of old and new convex formulations for the problem and its variants. We then review dual smoothing and level set techniques in convex optimization, present several novel theoretical results, and apply the techniques on the RPCA probl…

The paper classifies and constructs rotational surfaces with constant astigmatism in space forms.

Various problems in data analysis and statistical genetics call for recovery of a column-sparse, low-rank matrix from noisy observations. We propose ReFACTor, a simple variation of the classical Truncated Singular Value Decomposition (TSVD) algorithm. In contrast to previous sparse principal component analysis (PCA) al…

Let $C\to M$ be the bundle of connections of a principal bundle on $M$. The solutions to Hamilton-Cartan equations for a gauge-invariant Lagrangian density $Λ$ on $C$ satisfying a weak condition of regularity, are shown to admit an affine fibre-bundle structure over the set of solutions to Euler-Lagrange equations for …

Study of harmonic maps into principal bundles with applications to magnetic interactions.

Research in several fields now requires the analysis of data sets in which multiple high-dimensional types of data are available for a common set of objects. In particular, The Cancer Genome Atlas (TCGA) includes data from several diverse genomic technologies on the same cancerous tumor samples. In this paper we introd…

New stability theorem for hypersurfaces in Minkowski spaces.