arXiv research

Study optimizes estimation of orthogonal and rotation matrices from noisy data.

Study on rotating surfaces in 4D space with matrices.

A new method for sparse PCA using orthogonal rotations and soft-thresholding.

DFRot improves LLMs by reducing outlier and massive activation effects.

New quadratic forms expand and rotate linear endomorphisms in geometric theory.

Innovates rotation index for matrix pairs, solving group action problems.

Predict missing movie ratings or graph embeddings with low rank matrices.

Classifies SL(n) covariant matrix-valued valuations on Lp-spaces.

This note gives a simple analysis of a randomized approximation scheme for matrix multiplication proposed by Sarlos (2006) based on a random rotation followed by uniform column sampling. The result follows from a matrix version of Bernstein's inequality and a tail inequality for quadratic forms in subgaussian random ve…

Sparse principal component analysis (sparse PCA) aims at finding a sparse basis to improve the interpretability over the dense basis of PCA, meanwhile the sparse basis should cover the data subspace as much as possible. In contrast to most of existing work which deal with the problem by adding some sparsity penalties o…

The paper defines and calculates fourth fundamental form and i-th curvatures for hypersurfaces in 4D Euclidean space.

Rotates MFVI for better Gaussian approximations.

Paper proposes a new deflation varimax method for vintage factor analysis.

RP-GFRFT unifies fractional order and rotation control for graph signals.

Skein theory classifies UFCs with specific fusion rules.

Study connects covariance cleaning theory to information theory for heavy-tailed distributions.

Rotors were introduced in Graph Theory by W.Tutte. The concept was adapted to Knot Theory as a generalization of mutation by Anstee, Przytycki and Rolfsen in 1987. In this paper we show that Tristram-Levine signature is preserved by orientation-preserving rotations. Moreover, we show that any link invariant obtained fr…

Basis adaptation in Homogeneous Chaos spaces rely on a suitable rotation of the underlying Gaussian germ. Several rotations have been proposed in the literature resulting in adaptations with different convergence properties. In this paper we present a new adaptation mechanism that builds on compressive sensing algorith…

We investigate the problem of estimating a given real symmetric signal matrix $\textbf{C}$ from a noisy observation matrix $\textbf{M}$ in the limit of large dimension. We consider the case where the noisy measurement $\textbf{M}$ comes either from an arbitrary additive or multiplicative rotational invariant perturbati…

The paper develops a new algorithm for RBMs using dynamical mean-field theory.

Whitening, or sphering, is a common preprocessing step in statistical analysis to transform random variables to orthogonality. However, due to rotational freedom there are infinitely many possible whitening procedures. Consequently, there is a diverse range of sphering methods in use, for example based on principal com…

We consider probabilistic PCA and related factor models from a Bayesian perspective. These models are in general not identifiable as the likelihood has a rotational symmetry. This gives rise to complicated posterior distributions with continuous subspaces of equal density and thus hinders efficiency of inference as wel…

The paper classifies rotational hypersurfaces in n-space using a modified Laplacian operator.

Optimizing over the set of orthogonal matrices is a central component in problems like sparse-PCA or tensor decomposition. Unfortunately, such optimization is hard since simple operations on orthogonal matrices easily break orthogonality, and correcting orthogonality usually costs a large amount of computation. Here we…

We study $SU(2)$ calorons, also known as periodic instantons, and consider invariance under isometries of $S^1\times\mathbb{R}^3$ coupled with a non-spatial isometry called the rotation map. In particular, we investigate the fixed points under various cyclic symmetry groups. Our approach utilises a construction akin to…

OMD monitors stock market dynamics through matrix trajectories, revealing crisis patterns and sector rotations.

In (exploratory) factor analysis, the loading matrix is identified only up to orthogonal rotation. For identifiability, one thus often takes the loading matrix to be lower triangular with positive diagonal entries. In Bayesian inference, a standard practice is then to specify a prior under which the loadings are indepe…

OMD monitors stock market dynamics through matrix trajectories and reveals crisis patterns.

A new algorithm computes elastic shape distances between curves efficiently.

Bayes-optimal limits in PCA with structured noise are determined.

New algorithm reduces communication in distributed eigenspace estimation.

Gradient descent with noise converges to a unique optimum in nonconvex matrix factorization.

New method constructs equivariant neural networks for arbitrary matrix groups.

We propose a novel approach to addressing the vanishing (or exploding) gradient problem in deep neural networks. We construct a new architecture for deep neural networks where all layers (except the output layer) of the network are a combination of rotation, permutation, diagonal, and activation sublayers which are all…

We elaborate on the recent observation that evolution for twist knots simplifies when described in terms of triangular evolution matrix ${\cal B}$, not just its eigenvalues $Λ$, and provide a universal formula for ${\cal B}$, applicable to arbitrary rectangular representation $R=[r^s]$. This expression is in terms of s…

The concepts of unitary evolution matrices and associative memory have boosted the field of Recurrent Neural Networks (RNN) to state-of-the-art performance in a variety of sequential tasks. However, RNN still have a limited capacity to manipulate long-term memory. To bypass this weakness the most successful application…

MFVI mode collapse explained; RoVI proposed to mitigate.

Constructs coordinates to diagonalize Toda flow on matrices with simple spectrum.

Spectral clustering performance depends on eigenvector fluctuations, shown to be Gaussian.

Derives a sharp inequality for trace-free matrices with applications to hypersurfaces.

The paper studies dynamical systems with evolving geometric structure using numerical methods.

We have introduce a new vision of stochastic processes through the geometry induced by the dilation. The dilation matrices of a given processes are obtained by a composition of rotations matrices, contain the measure information in a condensed way. Particularly interesting is the fact that the obtention of dilation mat…

Subsampled Randomized Hadamard Transform (SRHT), a popular random projection method that can efficiently project a $d$-dimensional data into $r$-dimensional space ($r \ll d$) in $O(dlog(d))$ time, has been widely used to address the challenge of high-dimensionality in machine learning. SRHT works by rotating the input …

In the framework of geometric quantization we extend the Bohr-Sommerfeld rules to a full quantization theory which resembles Heisenberg's matrix theory. This extension is possible because Bohr-Sommerfeld rules not only provide an orthogonal basis in the space of quantum states, but also give a lattice structure to this…

New method proves asymptotic normality for matrix sensing problems.

A method for identifying joint and individual subspaces from multi-view data.

A new transform links rotating calorons to solutions of a differential equation.

We analyze the dynamics of an algorithm for approximate inference with large Gaussian latent variable models in a student-teacher scenario. To model nontrivial dependencies between the latent variables, we assume random covariance matrices drawn from rotation invariant ensembles. For the case of perfect data-model matc…