arXiv research

Subspace clustering is the problem of partitioning unlabeled data points into a number of clusters so that data points within one cluster lie approximately on a low-dimensional linear subspace. In many practical scenarios, the dimensionality of data points to be clustered are compressed due to constraints of measuremen…

We prove that for compact, non-contractible, one dimensional geodesic spaces, a version of the marked length spectrum conjecture holds. For a compact one dimensional geodesic space X, we define a subspace Conv(X). When X is non-contractible, we show that X deformation retracts to Conv(X). If two such spaces X, Y have t…

Method estimates multivariate counterfactual distributions efficiently and accurately.

Paper improves $\ell^{0}$-SSC for noisy data by proving SDP and proposing Noisy-DR-$\ell^{0}$-SSC.

Diffusion models improve creativity by smoothing the score function, leading to interpolated data.

The purpose of this paper is to classify $α$-para Kenmotsu manifolds $M^3$ such that the projection of the image of concircular curvature tensor $L$ in one-dimensional linear subspace of $T_{p}(M^{3})$ generated by $ξ_{p}$ is zero.

We propose a conjugate gradient type optimization technique for the computation of the Karcher mean on the set of complex linear subspaces of fixed dimension, modeled by the so-called Grassmannian. The identification of the Grassmannian with Hermitian projection matrices allows an accessible introduction of the geometr…

BOIDS optimizes high-dimensional problems by guiding optimization with one-dimensional lines.

Bayesian optimization is known to be difficult to scale to high dimensions, because the acquisition step requires solving a non-convex optimization problem in the same search space. In order to scale the method and keep its benefits, we propose an algorithm (LineBO) that restricts the problem to a sequence of iterative…

Proposes an optimization framework for sparse robust subspace estimation.

We study multi-moment maps induced by a two-torus action on the four homogeneous nearly Kähler six-manifolds. Their explicit expression and stationary orbits are derived. The configuration of fixed-points and one-dimensional orbits is worked out for generic six-manifolds equipped with an $\mathrm{SU}(3)$-structure admi…

Principal Component Analysis (PCA) is a method for estimating a subspace given noisy samples. It is useful in a variety of problems ranging from dimensionality reduction to anomaly detection and the visualization of high dimensional data. PCA performs well in the presence of moderate noise and even with missing data, b…

Studying a softmax-attention model, we show that the learned query converges to the latent signal subspace spanned by the informative direction.

New neural networks learn single-index models efficiently.

A new method slices and sums radial kernels faster.

Vector representations of words have heralded a transformational approach to classical problems in NLP; the most popular example is word2vec. However, a single vector does not suffice to model the polysemous nature of many (frequent) words, i.e., words with multiple meanings. In this paper, we propose a three-fold appr…

A new method for SVGD reduces variance in high dimensions.

Non-Gaussian component analysis (NGCA) is a problem in multidimensional data analysis which, since its formulation in 2006, has attracted considerable attention in statistics and machine learning. In this problem, we have a random variable $X$ in $n$-dimensional Euclidean space. There is an unknown subspace $Γ$ of the …

Defines and extends flat pseudo-Riemannian F-Lie algebras.

Max-sliced Wasserstein metric reduces high-dimensional data to 1D for better estimation.

Study one-dimensional topological theories with linear generating functions.

The paper explores properties of the Radon transform in relation to neural networks and ridges.

A new approach simplifies Sliced-Wasserstein distances to improve learning performance.

One popular trend in meta-learning is to learn from many training tasks a common initialization for a gradient-based method that can be used to solve a new task with few samples. The theory of meta-learning is still in its early stages, with several recent learning-theoretic analyses of methods such as Reptile [Nichol …

This work improves understanding of projection robust optimal transport distances.

We found a new simple family of Cantor sets whose projections are one-dimensional.

An Engel structure is a maximally non-integrable field of two-planes tangent to a four-manifold. Any two such structures are locally diffeomorphic. We investigate the space of global deformations of canonical Engel structures arising out of contact three-manifolds. The main tool is Cartan's method of prolongation and d…

Study of one-dimensional non-Hausdorff manifolds and their quotient to CW complexes.

A new metric-based principal curve method learns 1D manifolds from spatial data.

Reduces function approximation dimensions from high to low with sparse data.

One-dimensional crystals have convex shapes under certain conditions.

Quantum reservoir computing needs coherence influx for effective information processing.

Smooth algebra analysis for one-dimensional singular foliations.

We classify the harmonic morphisms with one-dimensional fibres (1) from real-analytic conformally-flat Riemannian manifolds of dimension at least four, and (2) between conformally-flat Riemannian manifolds of dimensions at least three.

Left invariant affine structures in a Lie group $G$ are in one-to-one correspondence with left-symmetric algebras over its Lie algebra $\mathfrak g=T_eG$ (``over'' means that the commutator $[x,y]=xy-yx$ coincides with the Lie bracket; left-symmetric algebras can be defined as Lie-admissible algebras such that the mult…

It is generally understood that a given one-dimensional diffusion may be transformed by Cameron-Martin-Girsanov measure change into another one-dimensional diffusion with the same volatility but a different drift. But to achieve this we have to know that the change-of-measure local martingale that we write down is a tr…

Paper bounds subspace estimator error from noisy projections.

PCA is one of the most widely used dimension reduction techniques. A related easier problem is "subspace learning" or "subspace estimation". Given relatively clean data, both are easily solved via singular value decomposition (SVD). The problem of subspace learning or PCA in the presence of outliers is called robust su…

Union of Subspaces (UoS) is a popular model to describe the underlying low-dimensional structure of data. The fine details of UoS structure can be described in terms of canonical angles (also known as principal angles) between subspaces, which is a well-known characterization for relative subspace positions. In this pa…

Study $Sp(n)$-orbits in complex and $Σ$-complex subspaces of Hermitian quaternionic vector spaces.

In subspace clustering, a group of data points belonging to a union of subspaces are assigned membership to their respective subspaces. This paper presents a new approach dubbed Innovation Pursuit (iPursuit) to the problem of subspace clustering using a new geometrical idea whereby subspaces are identified based on the…

We consider the problem of detecting whether a tensor signal having many missing entities lies within a given low dimensional Kronecker-Structured (KS) subspace. This is a matched subspace detection problem. Tensor matched subspace detection problem is more challenging because of the intertwined signal dimensions. We s…

Paper shows affine constraint is unnecessary for high-dimensional data.

A low-rank transformation learning framework for subspace clustering and classification is here proposed. Many high-dimensional data, such as face images and motion sequences, approximately lie in a union of low-dimensional subspaces. The corresponding subspace clustering problem has been extensively studied in the lit…

This paper investigates the generalization of Principal Component Analysis (PCA) to Riemannian manifolds. We first propose a new and general type of family of subspaces in manifolds that we call barycentric subspaces. They are implicitly defined as the locus of points which are weighted means of $k+1$ reference points.…

We obtain a deterministic characterisation of the \emph{no free lunch with vanishing risk}, the \emph{no generalised arbitrage} and the \emph{no relative arbitrage} conditions in the one-dimensional diffusion setting and examine how these notions of no-arbitrage relate to each other.

In this paper we continue our studies of the one dimensional conformal metric flows, which were introduced in [8]. In this part we mainly focus on evolution equations involving fourth order derivatives. The global existence and exponential convergence of metrics for the 1-Q and 4-Q flows are obtained.

Flow Matching models help generative models stay within the subspace of real data.