Related papers: Principal Graphs and Manifolds
Generalized matrix approximation plays a fundamental role in many machine learning problems, such as CUR decomposition, kernel approximation, and matrix low rank approximation. Especially with today's applications involved in larger and…
In the minimum planarization problem, given some $n$-vertex graph, the goal is to find a set of vertices of minimum cardinality whose removal leaves a planar graph. This is a fundamental problem in topological graph theory. We present a…
Molecule generation is central to a variety of applications. Current attention has been paid to approaching the generation task as subgraph prediction and assembling. Nevertheless, these methods usually rely on hand-crafted or external…
Big data mining is well known to be an important task for data science, because it can provide useful observations and new knowledge hidden in given large datasets. Proximity-based data analysis is particularly utilized in many real-life…
We present a mathematical and algorithmic scheme for learning the principal geometric elements in an image or 3D object. We build on recent work that convexifies the basic problem of finding a combination of a small number shapes that…
A $k$-truss is an edge-induced subgraph $H$ such that each of its edges belongs to at least $k-2$ triangles of $H$. This notion has been introduced around ten years ago in social network analysis and security, as a form of cohesive subgraph…
Analyzing high-dimensional data with manifold learning algorithms often requires searching for the nearest neighbors of all observations. This presents a computational bottleneck in statistical manifold learning when observations of…
We study graph partitioning problems from a min-max perspective, in which an input graph on n vertices should be partitioned into k parts, and the objective is to minimize the maximum number of edges leaving a single part. The two main…
Principal component analysis is a versatile tool to reduce dimensionality which has wide applications in statistics and machine learning. It is particularly useful for modeling data in high-dimensional scenarios where the number of…
In recent years it has become popular to study machine learning problems in a setting of ordinal distance information rather than numerical distance measurements. By ordinal distance information we refer to binary answers to distance…
We propose a framework of principal manifolds to model high-dimensional data. This framework is based on Sobolev spaces and designed to model data of any intrinsic dimension. It includes principal component analysis and principal curve…
We provide a general framework for getting expected linear time constant factor approximations (and in many cases FPTASs) to several well-known problems in Computational Geometry, such as $k$-center clustering and farthest nearest neighbor.…
Synthetic polymers are versatile and widely used materials. Similar to small organic molecules, a large chemical space of such materials is hypothetically accessible. Computational property prediction and virtual screening can accelerate…
Connected clustering denotes a family of constrained clustering problems in which we are given a distance metric and an undirected connectivity graph $G$ that can be completely unrelated to the metric. The aim is to partition the $n$…
Principal Component Analysis is a novel way of of dimensionality reduction. This problem essentially boils down to finding the top k eigen vectors of the data covariance matrix. A considerable amount of literature is found on algorithms…
Baker devised a powerful technique to obtain approximation schemes for various problems restricted to planar graphs. Her technique can be directly extended to various other graph classes, among the most general ones the graphs avoiding a…
Optimization problems, generalized equations, and the multitude of other variational problems invariably lead to the analysis of sets and set-valued mappings as well as their approximations. We review the central concept of set-convergence…
This article presents a general approximation-theoretic framework to analyze measure transport algorithms for probabilistic modeling. A primary motivating application for such algorithms is sampling -- a central task in statistical…
Finding a smallest subgraph that is k-edge-connected, or augmenting a k-edge-connected graph with a smallest subset of given candidate edges to become (k+1)-edge-connected, are among the most fundamental Network Design problems. They are…
Motivated by the analysis of high-dimensional neuroimaging signals located over the cortical surface, we introduce a novel Principal Component Analysis technique that can handle functional data located over a two-dimensional manifold. For…