Related papers: Machine Learning Friendly Set Version of Johnson-L…
In the recent paper [Jin, Kolda & Ward, arXiv:1909.04801], it is proved that the Kronecker fast Johnson-Lindenstrauss transform (KFJLT) is, in fact, a Johnson-Lindenstrauss transform, which had previously only been conjectured. In this…
The uncertainty quantification and error control of classifiers are crucial in many high-consequence decision-making scenarios. We propose a selective classification framework that provides an indecision option for any observations that…
Probabilistic programming and statistical computing are vibrant areas in the development of the Julia programming language, but the underlying infrastructure dramatically predates recent developments. The goal of MeasureTheory.jl is to…
How does the geometric representation of a dataset change after the application of each randomly initialized layer of a neural network? The celebrated Johnson--Lindenstrauss lemma answers this question for linear fully-connected neural…
A classical result of Johnson and Lindenstrauss states that a set of $n$ high dimensional data points can be projected down to $O(\log n/\epsilon^2)$ dimensions such that the square of their pairwise distances is preserved up to a small…
The Lovasz Local Lemma (LLL) is a probabilistic tool which has been used to show the existence of a variety of combinatorial structures with good "local" properties. The "LLL-distribution" can be used to show that the resulting structures…
This paper discusses the topic of dimensionality reduction for $k$-means clustering. We prove that any set of $n$ points in $d$ dimensions (rows in a matrix $A \in \RR^{n \times d}$) can be projected into $t = \Omega(k / \eps^2)$…
The Lov\'asz Local Lemma (LLL) is a very powerful tool in combinatorics and probability theory to show the possibility of avoiding all bad events under some weakly dependent conditions. In a seminal paper, Ambainis, Kempe, and Sattath (JACM…
Regression tree (RT) has been widely used in machine learning and data mining community. Given a target data for prediction, a regression tree is first constructed based on a training dataset before making prediction for each leaf node. In…
The Lopsided Lovasz Local Lemma (LLLL) is a cornerstone probabilistic tool for showing that it is possible to avoid a collection of "bad" events as long as their probabilities and interdependencies are sufficiently small. The strongest…
For any finite point set in $D$-dimensional space equipped with the 1-norm, we present random linear embeddings to $k$-dimensional space, with a new metric, having the following properties. For any pair of points from the point set that are…
In this paper we study constrained subspace approximation problem. Given a set of $n$ points $\{a_1,\ldots,a_n\}$ in $\mathbb{R}^d$, the goal of the {\em subspace approximation} problem is to find a $k$ dimensional subspace that best…
The problems of random projections and sparse reconstruction have much in common and individually received much attention. Surprisingly, until now they progressed in parallel and remained mostly separate. Here, we employ new tools from…
We prove a local central limit theorem (LCLT) for the number of points $N(J)$ in a region $J$ in $\mathbb R^d$ specified by a determinantal point process with an Hermitian kernel. The only assumption is that the variance of $N(J)$ tends to…
Inspired by the feedforward multilayer perceptron (FF-MLP), decision tree (DT) and extreme learning machine (ELM), a new classification model, called the subspace learning machine (SLM), is proposed in this work. SLM first identifies a…
Dimensionality reduction is an effective method for learning high-dimensional data, which can provide better understanding of decision boundaries in human-readable low-dimensional subspace. Linear methods, such as principal component…
Meta-learning synthesizes and leverages the knowledge from a given set of tasks to rapidly learn new tasks using very little data. Meta-learning of linear regression tasks, where the regressors lie in a low-dimensional subspace, is an…
Machine learning (ML) entered the field of computational micromagnetics only recently. The main objective of these new approaches is the automatization of solutions of parameter-dependent problems in micromagnetism such as fast response…
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…
Dimension reduction is the process of embedding high-dimensional data into a lower dimensional space to facilitate its analysis. In the Euclidean setting, one fundamental technique for dimension reduction is to apply a random linear map to…