arXiv research

Paper proposes a method to recover point configurations from noisy distance data.

Matrix profile has been recently proposed as a promising technique to the problem of all-pairs-similarity search on time series. Efficient algorithms have been proposed for computing it, e.g., STAMP, STOMP and SCRIMP++. All these algorithms use the z-normalized Euclidean distance to measure the distance between subsequ…

Paper tackles robust Euclidean distance estimation with sparse outliers.

APGD algorithm reconstructs point set from partial distance measurements.

Although recovering an Euclidean distance matrix from noisy observations is a common problem in practice, how well this could be done remains largely unknown. To fill in this void, we study a simple distance matrix estimate based upon the so-called regularized kernel estimate. We show that such an estimate can be chara…

Two algorithms estimate Wasserstein distance matrices from few entries for manifold learning.

We investigate some geometric properties of the real algebraic variety $Δ$ of symmetric matrices with repeated eigenvalues. We explicitly compute the volume of its intersection with the sphere and prove a Eckart-Young-Mirsky-type theorem for the distance function from a generic matrix to points in $Δ$. We exhibit conne…

The original k-means clustering method works only if the exact vectors representing the data points are known. Therefore calculating the distances from the centroids needs vector operations, since the average of abstract data points is undefined. Existing algorithms can be extended for those cases when the sole input i…

Unified framework for hyperbolic embeddings from mixed data types.

Classical multidimensional scaling only works well when the noisy distances observed in a high dimensional space can be faithfully represented by Euclidean distances in a low dimensional space. Advanced models such as Maximum Variance Unfolding (MVU) and Minimum Volume Embedding (MVE) use Semi-Definite Programming (SDP…

This paper proposes a representational model for grid cells. In this model, the 2D self-position of the agent is represented by a high-dimensional vector, and the 2D self-motion or displacement of the agent is represented by a matrix that transforms the vector. Each component of the vector is a unit or a cell. The mode…

Matrix Factorization is a popular non-convex optimization problem, for which alternating minimization schemes are mostly used. They usually suffer from the major drawback that the solution is biased towards one of the optimization variables. A remedy is non-alternating schemes. However, due to a lack of Lipschitz conti…

Python package for SPD matrix distances, reproducible and extensible.

WE constructs GP kernels for mixed inputs using weighted EDMs.

A fast binary embedding method preserves Euclidean distances in high-dimensional data.

Sharp bounds for max-sliced Wasserstein distances derived for empirical distributions.

We calculate Euclidean distance degrees for common manifold optimization types.

Efficiently implements MEG for low-rank matrix optimization problems.

Study infinite Euclidean distance discriminants of algebraic varieties.

Many interesting machine learning problems are best posed by considering instances that are distributions, or sample sets drawn from distributions. Previous work devoted to machine learning tasks with distributional inputs has done so through pairwise kernel evaluations between pdfs (or sample sets). While such an appr…

We report on experimental measurement of the Hilbert-Schmidt distance between two two-qubit states by many-particle interference. We demonstrate that our three-step method for measuring distances in Hilbert space is far less complex than reconstructing density matrices and that it can be applied in quantum-enhanced mac…

Algorithm reconstructs vertex positions in random geometric graphs with improved accuracy.

This paper corrects the proof of the Theorem 2 from the Gower's paper \cite[page 5]{Gower:1982} as well as corrects the Theorem 7 from Gower's paper \cite{Gower:1986}. The first correction is needed in order to establish the existence of the kernel function used commonly in the kernel trick e.g. for $k$-means clusterin…

The study compares Euclidean and cosine distances in medical drug prescription prediction.

How many samples are sufficient to guarantee that the eigenvectors and eigenvalues of the sample covariance matrix are close to those of the actual covariance matrix? For a wide family of distributions, including distributions with finite second moment and distributions supported in a centered Euclidean ball, we prove …

The paper extends manifold learning to arbitrary norms, improving molecular motion mapping.

Develops log-Euclidean Lie groups for SPD and correlation matrices.

Revisits Isomap, showing it constructs Euclidean representations of geodesic structure.

Extends manifold learning to non-Euclidean metrics.

Paper analyzes singular subspace estimation in noisy matrix models.

Euclidean nets reveal properties of higher-dimensional manifolds.

New Sliced-Wasserstein distances for non-Euclidean data.

The paper introduces a statistical distance matrix for better feature representation and clustering.

Proposes a new model to maximize out-of-sample Sharpe ratios by forecasting tangency portfolios.

We address noisy Euclidean distances in high dimensions, estimating noise levels and correcting distances.

Smooth maps preserve distances on specific revolution surfaces.

For time series comparisons, it has often been observed that z-score normalized Euclidean distances far outperform the unnormalized variant. In this paper we show that a z-score normalized, squared Euclidean Distance is, in fact, equal to a distance based on Pearson Correlation. This has profound impact on many distanc…

Non-negative matrix factorization (NMF) minimizes the Euclidean distance between the data matrix and its low rank approximation, and it fails when applied to corrupted data because the loss function is sensitive to outliers. In this paper, we propose a Truncated CauchyNMF loss that handle outliers by truncating large e…

New curvature concept preserves graph distances under operations.

Tensorized random projections reduce high-dimensional tensor size efficiently.

Introduces Grassmann Distance Complexity to measure algebraic set nearest point problems.

Representing graphs as sets of node embeddings in certain curved Riemannian manifolds has recently gained momentum in machine learning due to their desirable geometric inductive biases, e.g., hierarchical structures benefit from hyperbolic geometry. However, going beyond embedding spaces of constant sectional curvature…

Sparse random projection (RP) is a popular tool for dimensionality reduction that shows promising performance with low computational complexity. However, in the existing sparse RP matrices, the positions of non-zero entries are usually randomly selected. Although they adopt uniform sampling with replacement, due to lar…

The paper shows how to recover true node positions from a graph or similarity matrix.

We propose a novel Wasserstein method with a distillation mechanism, yielding joint learning of word embeddings and topics. The proposed method is based on the fact that the Euclidean distance between word embeddings may be employed as the underlying distance in the Wasserstein topic model. The word distributions of to…

This paper addresses the estimation of the latent dimensionality in nonnegative matrix factorization (NMF) with the β-divergence. The β-divergence is a family of cost functions that includes the squared Euclidean distance, Kullback-Leibler and Itakura-Saito divergences as special cases. Learning the model order is impo…

Study rigidity by logarithmic capacity and related functions.

A new method compares unaligned datasets using log-Euclidean signatures of SPD matrices.