arXiv subjects
Publications and source records attributed to Zoltan Szabo.
This paper generalizes the bordered-algebraic knot invariant introduced in an earlier paper, giving an invariant now with more algebraic structure. It also introduces signs to define these invariants with integral coefficients. We describe effective computations of the resulting invariant.
Random Fourier features (RFF) represent one of the most popular and wide-spread techniques in machine learning to scale up kernel algorithms. Despite the numerous successful applications of RFFs, unfortunately, quite little is understood theoretically on their optimality and limitations of their performance. Only recently, precise statistical-computational trade-offs have been established for RFFs in the approximation of kernel values, kernel ridge regression, kernel PCA and SVM classification. Our goal is to spark the investigation of optimality of RFF-based approximations in tasks involving not only function values but derivatives, which naturally lead to optimization problems with kernel derivatives. Particularly, in this paper, we focus on the approximation quality of RFFs for kernel derivatives and prove that the existing finite-sample guarantees can be improved exponentially in terms of the domain where they hold, using recent tools from unbounded empirical process theory. Our result implies that the same approximation guarantee is attainable for kernel derivatives using RFF as achieved for kernel values.
Maximum mean discrepancy (MMD), also called energy distance or N-distance in statistics and Hilbert-Schmidt independence criterion (HSIC), specifically distance covariance in statistics, are among the most popular and successful approaches to quantify the difference and independence of random variables, respectively. Thanks to their kernel-based foundations, MMD and HSIC are applicable on a wide variety of domains. Despite their tremendous success, quite little is known about when HSIC characterizes independence and when MMD with tensor product kernel can discriminate probability distributions. In this paper, we answer these questions by studying various notions of characteristic property of the tensor product kernel.
We define and study a bigraded knot invariant whose Euler characteristic is the Alexander polynomial, closely connected to knot Floer homology. The invariant is the homology of a chain complex whose generators correspond to Kauffman states for a knot diagram. The definition uses decompositions of knot diagrams: to a collection of points on the line, we associate a differential graded algebra; to a partial knot diagram, we associate modules over the algebra. The knot invariant is obtained from these modules by an appropriate tensor product.
We propose a novel adaptive test of goodness-of-fit, with computational cost linear in the number of samples. We learn the test features that best indicate the differences between observed samples and a reference model, by minimizing the false negative rate. These features are constructed via Stein's method, meaning that it is not necessary to compute the normalising constant of the model. We analyse the asymptotic Bahadur efficiency of the new test, and prove that under a mean-shift alternative, our test always has greater relative efficiency than a previous linear-time kernel test, regardless of the choice of parameters for that test. In experiments, the performance of our method exceeds that of the earlier linear-time test, and matches or exceeds the power of a quadratic-time kernel test. In high dimensions and where model structure may be exploited, our goodness of fit test performs far better than a quadratic-time two-sample test based on the Maximum Mean Discrepancy, with samples drawn from the model.
Knot Floer homology is an invariant for knots discovered by the authors and, independently, Jacob Rasmussen. The discovery of this invariant grew naturally out of studying how a certain three-manifold invariant, Heegaard Floer homology, changes as the three-manifold undergoes Dehn surgery along a knot. Since its original definition, thanks to the contributions of many researchers, knot Floer homology has emerged as a useful tool for studying knots in its own right. We give here a few selected highlights of this theory, and then move on to some new algebraic developments in the computation of knot Floer homology.
We modify the construction of knot Floer homology to produce a one-parameter family of homologies for knots in the three-sphere. These invariants can be used to give homomorphisms from the smooth concordance group to the integers, giving bounds on the four-ball genus and the concordance genus of knots. We give some applications of these homomorphisms.
Two semimetrics on probability distributions are proposed, given as the sum of differences of expectations of analytic functions evaluated at spatial or frequency locations (i.e, features). The features are chosen so as to maximize the distinguishability of the distributions, by optimizing a lower bound on test power for a statistical test using these features. The result is a parsimonious and interpretable indication of how and where two distributions differ locally. An empirical estimate of the test power criterion converges with increasing sample size, ensuring the quality of the returned features. In real-world benchmarks on high-dimensional text and image data, linear-time tests using the proposed semimetrics achieve comparable performance to the state-of-the-art quadratic-time maximum mean discrepancy test, while returning human-interpretable features that explain the test results.
We focus on the distribution regression problem: regressing to vector-valued outputs from probability measures. Many important machine learning and statistical tasks fit into this framework, including multi-instance learning and point estimation problems without analytical solution (such as hyperparameter or entropy estimation). Despite the large number of available heuristics in the literature, the inherent two-stage sampled nature of the problem makes the theoretical analysis quite challenging, since in practice only samples from sampled distributions are observable, and the estimates have to rely on similarities computed between sets of points. To the best of our knowledge, the only existing technique with consistency guarantees for distribution regression requires kernel density estimation as an intermediate step (which often performs poorly in practice), and the domain of the distributions to be compact Euclidean. In this paper, we study a simple, analytically computable, ridge regression-based alternative to distribution regression, where we embed the distributions to a reproducing kernel Hilbert space, and learn the regressor from the embeddings to the outputs. Our main contribution is to prove that this scheme is consistent in the two-stage sampled setup under mild conditions (on separable topological domains enriched with kernels): we present an exact computational-statistical efficiency trade-off analysis showing that our estimator is able to match the one-stage sampled minimax optimal rate [Caponnetto and De Vito, 2007; Steinwart et al., 2009]. This result answers a 17-year-old open question, establishing the consistency of the classical set kernel [Haussler, 1999; Gaertner et. al, 2002] in regression. We also cover consistency for more recent kernels on distributions, including those due to [Christmann and Steinwart, 2010].
A new computationally efficient dependence measure, and an adaptive statistical test of independence, are proposed. The dependence measure is the difference between analytic embeddings of the joint distribution and the product of the marginals, evaluated at a finite set of locations (features). These features are chosen so as to maximize a lower bound on the test power, resulting in a test that is data-efficient, and that runs in linear time (with respect to the sample size n). The optimized features can be interpreted as evidence to reject the null hypothesis, indicating regions in the joint domain where the joint distribution and the product of the marginals differ most. Consistency of the independence test is established, for an appropriate choice of features. In real-world benchmarks, independence tests using the optimized features perform comparably to the state-of-the-art quadratic-time HSIC test, and outperform competing O(n) and O(n log n) tests.
We study the relationship between Bar-Natan's perturbation in Khovanov homology and Szabo's geometric spectral sequence, and construct a link invariant that generalizes both into a common theory. We study a few properties of the new invariant, and introduce a family of s-invariants from the new theory in the same spirit as Rasmussen's s-invariant.
We introduce the Locally Linear Latent Variable Model (LL-LVM), a probabilistic model for non-linear manifold discovery that describes a joint distribution over observations, their manifold coordinates and locally linear maps conditioned on a set of neighbourhood relationships. The model allows straightforward variational optimisation of the posterior distribution on coordinates and locally linear maps from the latent space to the observation space given the data. Thus, the LL-LVM encapsulates the local-geometry preserving intuitions that underlie non-probabilistic methods such as locally linear embedding (LLE). Its probabilistic semantics make it easy to evaluate the quality of hypothesised neighbourhood relationships, select the intrinsic dimensionality of the manifold, construct out-of-sample extensions and to combine the manifold model with additional probabilistic models that capture the structure of coordinates within the manifold.
We propose Kernel Hamiltonian Monte Carlo (KMC), a gradient-free adaptive MCMC algorithm based on Hamiltonian Monte Carlo (HMC). On target densities where classical HMC is not an option due to intractable gradients, KMC adaptively learns the target's gradient structure by fitting an exponential family model in a Reproducing Kernel Hilbert Space. Computational costs are reduced by two novel efficient approximations to this gradient. While being asymptotically exact, KMC mimics HMC in terms of sampling efficiency, and offers substantial mixing improvements over state-of-the-art gradient free samplers. We support our claims with experimental studies on both toy and real-world applications, including Approximate Bayesian Computation and exact-approximate MCMC.
Kernel methods represent one of the most powerful tools in machine learning to tackle problems expressed in terms of function values and derivatives due to their capability to represent and model complex relations. While these methods show good versatility, they are computationally intensive and have poor scalability to large data as they require operations on Gram matrices. In order to mitigate this serious computational limitation, recently randomized constructions have been proposed in the literature, which allow the application of fast linear algorithms. Random Fourier features (RFF) are among the most popular and widely applied constructions: they provide an easily computable, low-dimensional feature representation for shift-invariant kernels. Despite the popularity of RFFs, very little is understood theoretically about their approximation quality. In this paper, we provide a detailed finite-sample theoretical analysis about the approximation quality of RFFs by (i) establishing optimal (in terms of the RFF dimension, and growing set size) performance guarantees in uniform norm, and (ii) presenting guarantees in $L^r$ ($1\le r<\infty$) norms. We also propose an RFF approximation to derivatives of a kernel with a theoretical study on its approximation quality.
In an earlier work, we introduced a family of t-modified knot Floer homologies, defined by modifying the construction of knot Floer homology HFK-minus. The resulting groups were then used to define concordance homomorphisms indexed by t in [0,2]. In the present work we elaborate on the special case t=1, and call the corresponding modified knot Floer homology the unoriented knot Floer homology. Using elementary methods (based on grid diagrams and normal forms for surface cobordisms), we show that the resulting concordance homomorphism gives a lower bound for the smooth 4-dimensional crosscap number of a knot K --- the minimal first Betti number of a smooth (possibly non-orientable) surface in the 4-disk that meets the boundary 3-sphere along the given knot K.
We focus on the distribution regression problem: regressing to a real-valued response from a probability distribution. Although there exist a large number of similarity measures between distributions, very little is known about their generalization performance in specific learning tasks. Learning problems formulated on distributions have an inherent two-stage sampled difficulty: in practice only samples from sampled distributions are observable, and one has to build an estimate on similarities computed between sets of points. To the best of our knowledge, the only existing method with consistency guarantees for distribution regression requires kernel density estimation as an intermediate step (which suffers from slow convergence issues in high dimensions), and the domain of the distributions to be compact Euclidean. In this paper, we provide theoretical guarantees for a remarkably simple algorithmic alternative to solve the distribution regression problem: embed the distributions to a reproducing kernel Hilbert space, and learn a ridge regressor from the embeddings to the outputs. Our main contribution is to prove the consistency of this technique in the two-stage sampled setting under mild conditions (on separable, topological domains endowed with kernels). For a given total number of observations, we derive convergence rates as an explicit function of the problem difficulty. As a special case, we answer a 15-year-old open question: we establish the consistency of the classical set kernel [Haussler, 1999; Gartner et. al, 2002] in regression, and cover more recent kernels on distributions, including those due to [Christmann and Steinwart, 2010].
We present ITE (information theoretical estimators) a free and open source, multi-platform, Matlab/Octave toolbox that is capable of estimating many different variants of entropy, mutual information, divergence, association measures, cross quantities, and kernels on distributions. Thanks to its highly modular design, ITE supports additionally (i) the combinations of the estimation techniques, (ii) the easy construction and embedding of novel information theoretical estimators, and (iii) their immediate application in information theoretical optimization problems. ITE also includes a prototype application in a central problem class of signal processing, independent subspace analysis and its extensions.
The aim of this paper is to introduce and study a geometric spectral sequence in Khovanov homology. The construction was motivated by a similar spectral sequence from Khovanov homology to Heegaard Floer homology.