Related papers: Manifoldron: Direct Space Partition via Manifold D…
Although many machine learning algorithms involve learning subspaces with particular characteristics, optimizing a parameter matrix that is constrained to represent a subspace can be challenging. One solution is to use Riemannian…
Understanding how neural systems efficiently process information through distributed representations is a fundamental challenge at the interface of neuroscience and machine learning. Recent approaches analyze the statistical and geometrical…
We consider problems of dimensionality reduction and learning data representations for continuous spaces with two or more independent degrees of freedom. Such problems occur, for example, when observing shapes with several components that…
Parameter space is not function space for neural network architectures. This fact, investigated as early as the 1990s under terms such as ``reverse engineering," or ``parameter identifiability", has led to the natural question of parameter…
Algorithms proposed for solving high-dimensional optimization problems with no derivative information frequently encounter the "curse of dimensionality," becoming ineffective as the dimension of the parameter space grows. One feature of a…
Manifold-learning techniques are routinely used in mining complex spatiotemporal data to extract useful, parsimonious data representations/parametrizations; these are, in turn, useful in nonlinear model identification tasks. We focus here…
Large scale dynamical systems (e.g. many nonlinear coupled differential equations) can often be summarized in terms of only a few state variables (a few equations), a trait that reduces complexity and facilitates exploration of behavioral…
System identification has greatly benefited from deep learning techniques, particularly for modeling complex, nonlinear dynamical systems with partially unknown physics where traditional approaches may not be feasible. However, deep…
We extend our study of Motion Planning via Manifold Samples (MMS), a general algorithmic framework that combines geometric methods for the exact and complete analysis of low-dimensional configuration spaces with sampling-based approaches…
Neural Networks (NNs) are the method of choice for building learning algorithms. Their popularity stems from their empirical success on several challenging learning problems. However, most scholars agree that a convincing theoretical…
Analyzing large volumes of high-dimensional data requires dimensionality reduction: finding meaningful low-dimensional structures hidden in their high-dimensional observations. Such practice is needed in atomistic simulations of complex…
We present a general and modular algorithmic framework for path planning of robots. Our framework combines geometric methods for exact and complete analysis of low-dimensional configuration spaces, together with practical, considerably…
Manifold learning is a popular and quickly-growing subfield of machine learning based on the assumption that one's observed data lie on a low-dimensional manifold embedded in a higher-dimensional space. This thesis presents a mathematical…
Data-driven modeling of human motions is ubiquitous in computer graphics and computer vision applications, such as synthesizing realistic motions or recognizing actions. Recent research has shown that such problems can be approached by…
Neural network pruning is an essential approach for reducing the computational complexity of deep models so that they can be well deployed on resource-limited devices. Compared with conventional methods, the recently developed dynamic…
We study the multiple manifold problem, a binary classification task modeled on applications in machine vision, in which a deep fully-connected neural network is trained to separate two low-dimensional submanifolds of the unit sphere. We…
Recurrent Neural Networks (RNNs) have found widespread applications in machine learning for time series prediction and dynamical systems reconstruction, and experienced a recent renaissance with improved training algorithms and…
We derive upper bounds on the complexity of ReLU neural networks approximating the solution maps of parametric partial differential equations. In particular, without any knowledge of its concrete shape, we use the inherent…
Deep neural networks have become the main work horse for many tasks involving learning from data in a variety of applications in Science and Engineering. Traditionally, the input to these networks lie in a vector space and the operations…
This paper proposes a novel paradigm for machine learning that moves beyond traditional parameter optimization. Unlike conventional approaches that search for optimal parameters within a fixed geometric space, our core idea is to treat the…