Selected papers: (Complete publication list is here.)
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Inverse Density as an Inverse Problem: The Fredholm Equation Approach
[arxiv]
Qichao Que, Mikhail Belkin
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+ abstract
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In this paper we address the problem of estimating the ratio $\frac{q}{p}$ where $p$ is a density function and $q$ is another density, or, more generally an arbitrary function. Knowing or approximating this ratio is needed in various problems of inference and integration, in particular, when one needs to average a function with respect to one probability distribution, given a sample from another. It is often referred as {\it importance sampling} in statistical inference and is also closely related to the problem of {\it covariate shift} in transfer learning as well as to various MCMC methods. It may also be useful for separating the underlying geometry of a space, say a manifold, from the density function defined on it.
Our approach is based on reformulating the problem of estimating $\frac{q}{p}$ as an inverse problem in terms of an integral operator corresponding to a kernel, and thus reducing it to an integral equation, known as the Fredholm problem of the first kind. This formulation, combined with the techniques of regularization and kernel methods, leads to a principled kernel-based framework for constructing algorithms and for analyzing them theoretically.
The resulting family of algorithms (FIRE, for Fredholm Inverse Regularized Estimator) is flexible, simple and easy to implement.
We provide detailed theoretical analysis including concentration bounds and convergence rates for the Gaussian kernel in the case of densities defined on $\R^d$, compact domains in $\R^d$ and smooth $d$-dimensional sub-manifolds of the Euclidean space.
We also show experimental results including applications to classification and semi-supervised learning within the covariate shift framework and demonstrate some encouraging experimental comparisons. We also show how the parameters of our algorithms can be chosen in a completely unsupervised manner.
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Blind Signal Separation in the Presence of Gaussian Noise
[arxiv]
Mikhail Belkin, Luis Rademacher, James Voss, COLT 2013
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+ abstract
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A prototypical blind signal separation problem is the so-called cocktail party problem, with n people talking simultaneously and n different microphones within a room. The goal is to recover each speech signal from the microphone inputs. Mathematically this can be modeled by assuming that we are given samples from a n-dimensional random variable X=AS, where S is a vector whose coordinates are independent random variables corresponding to each speaker. The objective is to recover the matrix A^{-1} given random samples from X. A range of techniques collectively known as Independent Component Analysis (ICA) have been proposed to address this problem in the signal processing and machine learning literature. Many of these techniques are based on using the kurtosis or other cumulants to recover the components.
In this paper we propose a new algorithm for solving the blind signal separation problem in the presence of additive Gaussian noise, when we are given samples from X=AS + \eta, where {\eta} is drawn from an unknown n-dimensional Gaussian distribution. Our approach is based on a method for decorrelating a sample with additive Gaussian noise under the assumption that the underlying distribution is a linear transformation of a distribution with independent components. Our decorrelation routine is based on the properties of cumulant tensors and can be combined with any standard cumulant-based method for ICA to get an algorithm that is provably robust in the presence of Gaussian noise. We derive polynomial bounds for sample complexity and error propagation of our method.
Our results generalize the recent work of Arora et al. which deals with a special case of ICA when S is the uniform probability distribution over the binary cube.
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Graph Laplacians on Singular Manifolds: Toward understanding complex spaces: graph Laplacians on manifolds with singularities and boundaries [arxiv]
Mikhail Belkin, Qichao Que, Yusu Wang, Xueyuan Zhou, COLT 2012.
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+ abstract
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Recently, much of the existing work in manifold learning has been done under the assumption that the data is sampled from a manifold without boundaries and singularities or that the functions of interest are evaluated away from such points. At the same time, it can be argued that singularities and boundaries are an important aspect of the geometry of realistic data.
In this paper we consider the behavior of graph Laplacians at points at or near boundaries and two main types of other singularities: intersections, where different manifolds come together and sharp "edges", where a manifold sharply changes direction. We show that the behavior of graph Laplacian near these singularities is quite different from that in the interior of the manifolds. In fact, a phenomenon somewhat reminiscent of the Gibbs effect in the analysis of Fourier series, can be observed in the behavior of graph Laplacian near such points. Unlike in the interior of the domain, where graph Laplacian converges to the Laplace-Beltrami operator, near singularities graph Laplacian tends to a first-order differential operator, which exhibits different scaling behavior as a function of the kernel width. One important implication is that while points near the singularities occupy only a small part of the total volume, the difference in scaling results in a disproportionately large contribution to the total behavior. Another significant finding is that while the scaling behavior of the operator is the same near different types of singularities, they are very distinct at a more refined level of analysis.
We believe that a comprehensive understanding of these structures in addition to the standard case of a smooth manifold can take us a long way toward better methods for analysis of complex non-linear data and can lead to significant progress in algorithm design.
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Data Skeletonization via Reeb Graphs
[pdf]
X. Ge, I. Safa, M. Belkin, Y. Wang, NIPS 2011.
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+ abstract
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Recovering hidden structure from complex and noisy non-linear data is one of the most fundamental problems in machine learning and statistical inference. While such data is often high-dimensional, it is of interest to approximate it with a low-dimensional or even one-dimensional space, since many important aspects of data are often intrinsically low-dimensional. Furthermore, there are many scenarios where the underlying structure is graph-like, e.g, river/road networks or various trajectories. In this paper, we develop a framework to extract, as well as to simplify, a one-dimensional "skeleton" from unorganized data using the Reeb graph. Our algorithm is very simple, does not require complex optimizations and can be easily applied to unorganized high-dimensional data such as point clouds or proximity graphs. It can also represent arbitrary graph structures in the data. We also give theoretical results to justify our method. We provide a number of experiments to demonstrate the effectiveness and generality of our algorithm, including comparisons to existing methods, such as principal curves. We believe that the simplicity and practicality of our algorithm will help to promote skeleton graphs as a data analysis tool for a broad range of applications.
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Polynomial Learning of Distribution Families [arxiv]
M. Belkin, K. Sinha, FOCS 2010.
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+ abstract
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The question of polynomial learnability of probability distributions, particularly Gaussian mixture distributions, has recently received significant attention in theoretical computer science and machine learning. However, despite major progress, the general question of polynomial learnability of Gaussian mixture distributions still remained open. The current work resolves the question of polynomial learnability for Gaussian mixtures in high dimension with an arbitrary fixed number of components. The result on learning Gaussian mixtures relies on an analysis of distributions belonging to what we call "polynomial families" in low dimension. These families are characterized by their moments being polynomial in parameters and include almost all common probability distributions as well as their mixtures and products. Using tools from real algebraic geometry, we show that parameters of any distribution belonging to such a family can be learned in polynomial time and using a polynomial number of sample points. The result on learning polynomial families is quite general and is of independent interest. To estimate parameters of a Gaussian mixture distribution in high dimensions, we provide a deterministic algorithm for dimensionality reduction. This allows us to reduce learning a high-dimensional mixture to a polynomial number of parameter estimations in low dimension. Combining this reduction with the results on polynomial families yields our result on learning arbitrary Gaussian mixtures in high dimensions.
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Convergence of Laplacian Eigenmaps [pdf, bib]
M. Belkin, P. Niyogi
preprint, short version NIPS 2008.
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+ abstract
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Geometrically based methods for various tasks of data analysis
have attracted considerable attention over the last few years.
In many of these algorithms, a central role is played by the
eigenvectors of the graph Laplacian of a
data-derived graph. In this paper, we show that if points are
sampled uniformly at random from an unknown submanifold ${\cal M}$
of $\R^N$, then the eigenvectors of a suitably constructed
graph Laplacian converge to the eigenfunctions of the Laplace
Beltrami operator on ${\cal M}$. This basic result directly
establishes the convergence of spectral manifold learning
algorithms such as Laplacian Eigenmaps and Diffusion Maps.
It also has implications for the understanding of geometric algorithms
in data analysis, computational harmonic analysis, geometric
random graphs, and graphics.
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On Learning with Integral Operators
[pdf, bib]
L. Rosasco, M.Belkin, E. De Vito,
Journal of Machine Learning Research, vol.11, pp.905-934, 2010.
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+ abstract
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A large number of learning algorithms, for example,
spectral clustering, kernel Principal Components
Analysis and many manifold methods are
based on estimating eigenvalues and eigenfunctions
of operators defined by a similarity function or a
kernel, given empirical data. Thus for the analysis
of algorithms, it is an important problem to
be able to assess the quality of such approximations.
The contribution of our paper is two-fold.
First, we use a technique based on a concentration
inequality for Hilbert spaces to provide new simplified
proofs for a number of results in spectral
approximation. Second, using these methods we
provide several new results for estimating spectral
properties of the graph Laplacian operator extending
and strengthening results from [28].
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Data spectroscopy: eigenspaces of convolution operators and clustering [pdf, bib]
Tao Shi, Mikhail Belkin, Bin Yu
The Annals of Statistics, vol. 37, Number 6B (2009), 3960-3984.
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+ abstract
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This paper focuses on obtaining clustering information about a
distribution from its i.i.d. samples. We develop theoretical results to
understand and use clustering information contained in the eigenvectors
of data adjacency matrices based on a radial kernel function
with a sufficiently fast tail decay. In particular, we provide population
analyses to gain insights into which eigenvectors should be used and
when the clustering information for the distribution can be recovered
from the sample. We learn that a fixed number of top eigenvectors
might at the same time contain redundant clustering information and
miss relevant clustering information. We use this insight to design
the Data Spectroscopic clustering (DaSpec) algorithm that utilizes
properly selected eigenvectors to determine the number of clusters
automatically and to group the data accordingly. Our findings extend
the intuitions underlying existing spectral techniques such as
spectral clustering and Kernel Principal Components Analysis, and
provide new understanding into their usability and modes of failure.
Simulation studies and experiments on real world data are conducted
to show the potential of our algorithm. In particular, DaSpec is found
to handle unbalanced groups and recover clusters of different shapes
better than the competing methods.
- Towards a Theoretical Foundation for Laplacian-Based
Manifold Methods [pdf, bib]
M. Belkin, P. Niyogi
Journal of Computer and System Sciences, 2008.
Volume 74, Issue 8, pp. 1289-1308.
Special Issue on Learning Theory, invited (short version in COLT 2005).
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+ abstract
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In recent years manifold methods have attracted a considerable amount of atten-
tion in machine learning. However most algorithms in that class may be termed
"manifold-motivated" as they lack any explicit theoretical guarantees. In this pa-
per we take a step towards closing the gap between theory and practice for a class
of Laplacian-based manifold methods. These methods utilize the graph Laplacian
associated to a data set for a variety of applications in semi-supervised learning,
clustering, data representation.
We show that under certain conditions the graph Laplacian of a point cloud of
data samples converges to the Laplace-Beltrami operator on the underlying mani-
fold. Theorem 3.1 contains the .rst result showing convergence of a random graph
Laplacian to the manifold Laplacian in the context of machine learning.
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Discrete Laplace Operator for Meshed Surfaces
[pdf, code, bib]
M. Belkin, J. Sun, Y. Wang, 24th Annual Symposium on Computational Geometry (SOCG) 2008.
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+ abstract
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In recent years a considerable amount of work in graphics and geometric optimization used tools
based on the Laplace-Beltrami operator on a surface. The applications of the Laplacian include mesh
editing, surface smoothing, and shape interpolations among others. However, it has been shown [12, 23,
25] that the popular cotangent approximation schemes do not provide convergent point-wise (or even
L^2) estimates, while many applications rely on point-wise estimation. Existence of such schemes has
been an open question [12].
In this paper we propose the first algorithm for approximating the Laplace operator of a surface from
a mesh with point-wise convergence guarantees applicable to arbitrary meshed surfaces. We show that
for a sufficiently fine mesh over an arbitrary surface, our mesh Laplacian is close to the Laplace-Beltrami
operator on the surface at every point of the surface.
Moreover, the proposed algorithm is simple and easily implementable. Experimental evidence shows
that our algorithm exhibits convergence empirically and outperforms cotangent-based methods in providing
accurate approximation of the Laplace operator for various meshes.
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Consistency of Spectral Clustering [pdf, bib]
U. von Luxburg, M. Belkin, O. Bousquet,
The Annals of Statistics 2008, Vol. 36, No. 2, 555-586.
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+ abstract
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Consistency is a key property of statistical
algorithms when the data is drawn from some
underlying probability distribution. Surprisingly,
despite decades of work, little is known about
consistency of most clustering algorithms. In this
paper we investigate consistency of the popular
family of spectral clustering algorithms, which
clusters the data with the help of eigenvectors of
graph Laplacian matrices. We develop new methods to
establish that for increasing sample size, those
eigenvectors converge to the eigenvectors of certain
limit operators. As a result we can prove that one of
the two major classes of spectral clustering
(normalized clustering) converges under very general
conditions, while the other (unnormalized clustering)
is only consistent under strong additional
assumptions, which are not always satisfied in real
data. We conclude that our analysis provides strong
evidence for the superiority of normalized spectral
clustering.
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Manifold Regularization:
a Geometric Framework for Learning from Labeled and
Unlabeled Examples [pdf, bib]
M. Belkin, P. Niyogi, V. Sindhwani
Journal of Machine Learning Research, 7(Nov):2399-2434,
2006.
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+ abstract
- We propose a family of learning algorithms based on a
new form of regularization that allows us to exploit
the geometry of the marginal distribution. We focus
on a semi-supervised framework that incorporates
labeled and unlabeled data in a general-purpose
learner. Some transductive graph learning algorithms
and standard methods including support vector
machines and regularized least squares can be
obtained as special cases. We use properties of
reproducing kernel Hilbert spaces to prove new
Representer theorems that provide theoretical basis
for the algorithms. As a result (in contrast to
purely graph-based approaches) we obtain a natural
out-of-sample extension to novel examples and so are
able to handle both transductive and truly
semi-supervised settings. We present experimental
evidence suggesting that our semi-supervised
algorithms are able to use unlabeled data
effectively. Finally we have a brief discussion of
unsupervised and fully supervised learning within our
general framework.
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Heat Flow and a Faster Algorithm to Compute the
Surface Area of a Convex Body [pdf, bib]
M. Belkin, H. Narayanan, P. Niyogi, FOCS 2006.
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+ abstract
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We draw on the observation that the amount of heat
diffusing outside of a heated body in a short period
of time is proportional to its surface area, to
design a simple algorithm for approximating the
surface area of a convex body given by a membership
oracle. Our method has a complexity of O*(n^4), where
n is the dimension, compared to O*(n^8.5) for the
previous best algorithm. We show that our complexity
cannot be improved given the current state-of-the-art
in volume estimation.
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Semi-supervised Learning on Riemannian Manifolds
[pdf, bib]
M. Belkin, P. Niyogi
Machine Learning, 56 (invited, special Issue on
clustering), 209-239, 2004 (short version in NIPS 2002).
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+ abstract
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We consider the general problem of utilizing both
labeled and unlabeled data to improve classification
accuracy. Under the assumption that the data lie on a
submanifold in a high dimensional space, we develop
an algorithmic framework to classify a partially
labeled data set in a principled manner. The central
idea of our approach is that classification functions
are naturally defined only on the submanifold in
question rather than the total ambient space. Using
the Laplace-Beltrami operator one produces a basis
(the Laplacian Eigenmaps) for a Hilbert space of
square integrable functions on the submanifold. To
recover such a basis, only unlabeled examples are
required. Once such a basis is obtained, training can
be performed using the labeled data set. Our
algorithm models the manifold using the adjacency
graph for the data and approximates the Laplace-
Beltrami operator by the graph Laplacian. We provide
details of the algorithm, its theoretical
justification, and several practical applications for
image, speech, and text classification.
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Laplacian Eigenmaps for Dimensionality Reduction and
Data Representation [pdf, bib]
M. Belkin, P. Niyogi
Neural Computation, June 2003; 15 (6):1373-1396 (short version in NIPS 2001).
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+ abstract
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One of the central problems in machine learning and
pattern recognition is to develop appropriate
representations for complex data. We consider the
problem of constructing a representation for data
lying on a lowdimensional manifold embedded in a
high-dimensional space. Drawing on the correspondence
between the graph Laplacian, the Laplace Beltrami
operator on the manifold, and the connections to the
heat equation, we propose a geometrically motivated
algorithm for representing the highdimensional data.
The algorithm provides a computationally efficient
approach to non
dimensionality reduction that
has locality-preserving properties and a natural
connection to clustering. Some potential applications
and illustrative examples are discussed.
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