Related papers: Learning on manifolds without manifold learning
We address the problem of approximating an unknown function from its discrete samples given at arbitrarily scattered sites. This problem is essential in numerical sciences, where modern applications also highlight the need for a solution to…
The manifold hypothesis, which assumes that data lies on or close to an unknown manifold of low intrinsic dimension, is a staple of modern machine learning research. However, recent work has shown that real-world data exhibits distinct…
There has been a growing interest in statistical inference from data satisfying the so-called manifold hypothesis, assuming data points in the high-dimensional ambient space to lie in close vicinity of a submanifold of much lower dimension.…
Motivated by the problem of tuning hyperparameters in machine learning, we present a new approach for gradually and adaptively optimizing an unknown function using estimated gradients. We validate the empirical performance of the proposed…
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…
A central goal of unsupervised learning is to acquire representations from unlabeled data or experience that can be used for more effective learning of downstream tasks from modest amounts of labeled data. Many prior unsupervised learning…
We study a regression problem on a compact manifold M. In order to take advantage of the underlying geometry and topology of the data, the regression task is performed on the basis of the first several eigenfunctions of the Laplace-Beltrami…
Classifiers label data as belonging to one of a set of groups based on input features. It is challenging to obtain accurate classification performance when the feature distributions in the different classes are complex, with nonlinear,…
Kernel density estimation is a key component of a wide variety of algorithms in machine learning, Bayesian inference, stochastic dynamics and signal processing. However, the unsupervised density estimation technique requires tuning a…
Traditional manifold learning algorithms assumed that the embedded manifold is globally or locally isometric to Euclidean space. Under this assumption, they divided manifold into a set of overlapping local patches which are locally…
Many statistical and machine learning approaches rely on pairwise distances between data points. The choice of distance metric has a fundamental impact on performance of these procedures, raising questions about how to appropriately…
Subspace clustering is an important unsupervised clustering approach. It is based on the assumption that the high-dimensional data points are approximately distributed around several low-dimensional linear subspaces. The majority of the…
Measuring the similarity between data points often requires domain knowledge, which can in parts be compensated by relying on unsupervised methods such as latent-variable models, where similarity/distance is estimated in a more compact…
We study the problem of learning local metrics for nearest neighbor classification. Most previous works on local metric learning learn a number of local unrelated metrics. While this "independence" approach delivers an increased flexibility…
I present a strategy for unsupervised manifold learning on local atomic environments in molecular simulations based on simple rotation- and permutation-invariant three-body features. These features are highly descriptive, generalize to…
We consider the problem of approximating a subset $M$ of a Hilbert space $X$ by a low-dimensional manifold $M_n$, using samples from $M$. We propose a nonlinear approximation method where $M_n $ is defined as the range of a smooth nonlinear…
Recently, there has been much interest in spectral approaches to learning manifolds---so-called kernel eigenmap methods. These methods have had some successes, but their applicability is limited because they are not robust to noise. To…
Manifold regularization is a commonly used technique in semi-supervised learning. It enforces the classification rule to be smooth with respect to the data-manifold. Here, we derive sample complexity bounds based on pseudo-dimension for…
We consider the problem of learning a manifold from a teacher's demonstration. Extending existing approaches of learning from randomly sampled data points, we consider contexts where data may be chosen by a teacher. We analyze learning from…
For challenging machine learning problems such as zero-shot learning and fine-grained categorization, embedding learning is the machinery of choice because of its ability to learn generic notions of similarity, as opposed to class-specific…