arXiv research

Two-dimensional embeddings remain the dominant approach to visualize high dimensional data. The choice of embeddings ranges from highly non-linear ones, which can capture complex relationships but are difficult to interpret quantitatively, to axis-aligned projections, which are easy to interpret but are limited to biva…

Investigates projections onto explicit subspaces and their variance effects.

This paper surveys various methods for dimensionality reduction and nearest neighbor search.

Study shows double descent curve in high-dimensional linear regression with random projections.

New insights into continual learning for deep models, showing convergence issues but local linear solutions.

We introduce switched linear projections for expressing the activity of a neuron in a deep neural network in terms of a single linear projection in the input space. The method works by isolating the active subnetwork, a series of linear transformations, that determine the entire computation of the network for a given i…

We present a projectively invariant description of planar linear 3-webs. For a non-hexagonal 3-web, we introduce family of projective torsion-free Cartan connections, the web leaves being geodesics for each member of the family, and give a web linearization criterion. Finally, we propose an algorithm for resolving the …

Geometrically transforms nonconservative dynamics to linearize Kepler and Manev systems.

Paper studies PSGD for constrained optimization problems and its statistical properties.

Computes $L_\infty$-algebroid for linear foliations on vector spaces.

Study Lie algebroid connections on principal bundles over complex projective varieties.

The paper classifies and decomposes quaternionic projective transformations.

We propose a geometric correspondence between (a) linearly degenerate systems of conservation laws with rectilinear rarefaction curves and (b) congruences of lines in projective space whose developable surfaces are planar pencils of lines. We prove that in projective 4-space such congruences are necessarily linear. Bas…

The paper analyzes an ensemble of randomly projected linear discriminants for high-dimensional data.

A special linear Lie group over the real number field and the quarternion field admits a projectivley flat affine connection. We show that parabolic subgroups are autoparallel submanifolds and give a criterion the induced connection is projectively equivalent to a flat affine connection.

H. Sato introduced a Schwarzian derivative of a contactomorphism of three-dimensional Euclidean space and with T. Ozawa described its basic properties. In this note their construction is extended to all odd dimensions and to non-flat contact projective structures. The contact projective Schwarzian derivative of a conta…

Study calculates Kulkarni limit sets for quaternionic projective groups.

For compact CR manifolds of hypersurface type which embed in complex projective space, we show that for all k large enough there exist linear systems of ${\mathcal{O}}(k)$ which when restricted to the CR manifold are generic in a suitable sense. These systems are constructed using approximately holomorphic geometry.

In a linear stochastic bandit model, each arm is a vector in an Euclidean space and the observed return at each time step is an unknown linear function of the chosen arm at that time step. In this paper, we investigate the problem of learning the best arm in a linear stochastic bandit model, where each arm's expected r…

Efficiently projects points onto polytopes, especially useful in web-scale applications.

In many online learning problems the computational bottleneck for gradient-based methods is the projection operation. For this reason, in many problems the most efficient algorithms are based on the Frank-Wolfe method, which replaces projections by linear optimization. In the general case, however, online projection-fr…

Large sectors of the recent optimization literature focused in the last decade on the development of optimal stochastic first order schemes for constrained convex models under progressively relaxed assumptions. Stochastic proximal point is an iterative scheme born from the adaptation of proximal point algorithm to nois…

Neural networks use their hidden layers to transform input data into linearly separable data clusters, with a linear or a perceptron type output layer making the final projection on the line perpendicular to the discriminating hyperplane. For complex data with multimodal distributions this transformation is difficult t…

Here, a non-linear analysis method is applied rather than classical one to study projective changes of Finsler metrics. More intuitively, a projectively invariant pseudo-distance is introduced and characterized with respect to the Ricci tensor and its covariant derivatives.

New method converts LVAs into linear projections for better understanding of complex models.

LightOn OPUs accelerate randomized numerical linear algebra, reducing computational costs.

The equation determining whether a projective structure admits a connection in its given projective class that has skew-symmetric Ricci tensor is an overdetermined system of semi-linear partial differential equations which we call the projective Einstein-Weyl (pEW) equation. In 2-dimensions, we give local obstructions …

MILDA uses unlabelled data to compute LDA projections.

The study provides a criterion for fractional-linear integrals of geodesics on surfaces.

Projection-cost preservation is a low-rank approximation guarantee which ensures that the cost of any rank-$k$ projection can be preserved using a smaller sketch of the original data matrix. We present a general structural result outlining four sufficient conditions to achieve projection-cost preservation. These condit…

A new method optimizes projection directions for sliced Wasserstein distances.

In his celebrated paper "Generic projections", John Mather has shown that almost all linear projections from a submanifold of a vector space into a subspace are transverse with respect to a given modular submanifold. In this paper, an improvement of Mather's result is stated. Namely, we show that almost all linear pert…

The approximation of nonlinear kernels via linear feature maps has recently gained interest due to their applications in reducing the training and testing time of kernel-based learning algorithms. Current random projection methods avoid the curse of dimensionality by embedding the nonlinear feature space into a low dim…

We propose a twistor construction of surfaces in Lie sphere geometry based on the linear system which copies equations of Wilczynski's projective frame. In the particular case of Lie-applicable surfaces this linear system describes joint eigenfunctions of a pair of commuting Schrödinger operators with magnetic fields.

Classifies geodesic flows on projective plane with potential field.

We study the affine quasi-Einstein equation, a second order linear homogeneous equation, which is invariantly defined on any affine manifold. We prove that the space of solutions is finite-dimensional, and its dimension is a strongly projective invariant. Moreover the maximal dimension is shown to be achieved if and on…

A point $p\in\mathbb{P}^N$ of a projective space is $h$-identifiable, with respect to a variety $X\subset\mathbb{P}^N$, if it can be written as linear combination of $h$ elements of $X$ in a unique way. Identifiability is implied by conditions on the contact locus in $X$ of general linear spaces called non weak defecti…

We present function preserving projections (FPP), a scalable linear projection technique for discovering interpretable relationships in high-dimensional data. Conventional dimension reduction methods aim to maximally preserve the global and/or local geometric structure of a dataset. However, in practice one is often mo…

Paper tackles online learning on curved spaces without projections.

We investigate contact magnetic curves in the real special linear group of degree 2. They are geodesics of the Hopf tubes over the projection curve. We prove that periodic contact magnetic curves in SL(2,R) can be quantized in the set of rational numbers. Finally, we study contact homogeneous magnetic trajectories in S…

In this paper, we propose and study random maxout features, which are constructed by first projecting the input data onto sets of randomly generated vectors with Gaussian elements, and then outputing the maximum projection value for each set. We show that the resulting random feature map, when used in conjunction with …

We extend the notion of a Thomas projective connection (a projective equivalence class of linear connections) for supermanifolds. As a by-product, we arrive at a generalisation of the multidimensional Schwarzian derivative for the super case which was previously unknown. This is combined with our previous construction …

The $\mathcal D$-groupoid of symmetries is minimal under specific conditions.

This paper presents a new ensemble learning method for classification problems called projection pursuit random forest (PPF). PPF uses the PPtree algorithm introduced in Lee et al. (2013). In PPF, trees are constructed by splitting on linear combinations of randomly chosen variables. Projection pursuit is used to choos…

The paper sharpens the analysis of sketch-and-project methods using randomized singular value decomposition.

Linearized Einstein equations simplified via Calabi operator.

We give a characterization of the boundaries of holomorphic chains in complex projective space in terms of certain non-linear moment conditions. This extends previous work of the authors and complements results of Dolbeault and Henkin.

New explanation of reservoir computing using random projections.