Predicting the nodes of a given graph is a fascinating theoretical problem with applications in several domains. Since graph sparsification via spanning trees retains enough information while making the task much easier, trees are an important special case of this problem. Although it is known how to predict the nodes of an unweighted tree in a nearly optimal way, in the weighted case a fully satisfactory algorithm is not available yet. We fill this hole and introduce an efficient node predictor, Shazoo, which is nearly optimal on any weighted tree. Moreover, we show that Shazoo can be viewed as a common nontrivial generalization of both previous approaches for unweighted trees and weighted lines. Experiments on real-world datasets confirm that Shazoo performs well in that it fully exploits the structure of the input tree, and gets very close to (and sometimes better than) less scalable energy minimization methods.

In this paper, we give a new generalization error bound of Multiple Kernel Learning (MKL) for a general class of regularizations. Our main target in this paper is dense type regularizations including ℓp-MKL that imposes ℓp-mixed-norm regularization instead of ℓ1-mixed-norm regularization. According to the recent numerical experiments, the sparse regularization does not necessarily show a good performance compared with dense type regularizations. Motivated by this fact, this paper gives a general theoretical tool to derive fast learning rates that is applicable to arbitrary monotone norm-type regularizations in a unifying manner. As a by-product of our general result, we show a fast learning rate of ℓp-MKL that is tightest among existing bounds. We also show that our general learning rate achieves the minimax lower bound. Finally, we show that, when the complexities of candidate reproducing kernel Hilbert spaces are inhomogeneous, dense type regularization shows better learning rate compared with sparse ℓ1 regularization.

Vector Auto-regressive models (VAR) are useful tools for analyzing time series data. In quite a few modern time series modelling tasks, the collection of reliable time series turns out to be a major challenge, either due to the slow progression of the dynamic process of interest, or inaccessibility of repetitive measurements of the same dynamic process over time. In those situations, however, we observe that it is often easier to collect a large amount of non-sequence samples, or snapshots of the dynamic process of interest. In this work, we assume a small amount of time series data are available, and propose methods to incorporate non-sequence data into penalized least-square estimation of VAR models. We consider non-sequence data as samples drawn from the stationary distribution of the underlying VAR model, and devise a novel penalization scheme based on the discrete-time Lyapunov equation concerning the covariance of the stationary distribution. Experiments on synthetic and video data demonstrate the effectiveness of the proposed methods.

We prove a new oracle inequality for support vector machines with Gaussian RBF kernels solving the regularized least squares regression problem. To this end, we apply the modulus of smoothness. With the help of the new oracle inequality we then derive learning rates that can also be achieved by a simple data-dependent parameter selection method. Finally, it turns out that our learning rates are asymptotically optimal for regression functions satisfying certain standard smoothness conditions.

For many of the state-of-the-art computer vision algorithms, image segmentation is an important preprocessing step. As such, several image segmentation algorithms have been proposed, however, with certain reservation due to high computational load and many hand-tuning parameters. Correlation clustering, a graph-partitioning algorithm often used in natural language processing and document clustering, has the potential to perform better than previously proposed image segmentation algorithms. We improve the basic correlation clustering formulation by taking into account higher-order cluster relationships. This improves clustering in the presence of local boundary ambiguities. We first apply the pairwise correlation clustering to image segmentation over a pairwise superpixel graph and then develop higher-order correlation clustering over a hypergraph that considers higher-order relations among superpixels. Fast inference is possible by linear programming relaxation, and also effective parameter learning framework by structured support vector machine is possible. Experimental results on various datasets show that the proposed higher-order correlation clustering outperforms other state-of-the-art image segmentation algorithms.

While loopy Belief Propagation (LBP) has been utilized in a wide variety of applications with empirical success, it comes with few theoretical guarantees. Especially, if the interactions of random variables in a graphical model are strong, the behaviors of the algorithm can be difficult to analyze due to underlying phase transitions. In this paper, we develop a novel approach to the uniqueness problem of the LBP fixed point; our new “necessary and sufficient” condition is stated in terms of graphs and signs, where the sign denotes the types (attractive/repulsive) of the interaction (i.e., compatibility function) on the edge. In all previous works, uniqueness is guaranteed only in the situations where the strength of the interactions are “sufficiently” small in certain senses. In contrast, our condition covers arbitrary strong interactions on the specified class of signed graphs. The result of this paper is based on the recent theoretical advance in the LBP algorithm; the connection with the graph zeta function.

In this paper we present an algorithm to learn a multi-label classifier which attempts at directly optimising the F-score. The key novelty of our formulation is that we explicitly allow for assortative (submodular) pairwise label interactions, i.e., we can leverage the co-ocurrence of pairs of labels in order to improve the quality of prediction. Prediction in this model consists of minimising a particular submodular set function, what can be accomplished exactly and efficiently via graph-cuts. Learning however is substantially more involved and requires the solution of an intractable combinatorial optimisation problem. We present an approximate algorithm for this problem and prove that it is sound in the sense that it never predicts incorrect labels. We also present a nontrivial test of a sufficient condition for our algorithm to have found an optimal solution. We present experiments on benchmark multi-label datasets, which attest the value of our proposed technique. We also make available source code that enables the reproduction of our experiments.

The distance dependent Chinese restaurant process (ddCRP) was recently introduced toaccommodate random partitions of non-exchangeable data [1]. The dd-CRP clusters data in a biased way: each data point is more likely to be clustered with other data that are near it in an external sense. This paper examines the dd-CRP in a spatial setting with the goal of natural image segmentation. We explore the biases of the spatial ddCRP model and propose a novel hierarchical extension bettersuited for producing "humanlike" segmentations. We then study the sensitivity of the models to various distance and appearance hyperparameters, and provide the first rigorous comparison of nonparametric Bayesian models in the image segmentationdomain. On unsupervised image segmentation, we demonstrate that similar performance to existing nonparametric Bayesian models is possible with substantially simpler models and algorithms.

We consider the problem of optimizing the sum of a smooth convex function and a non-smooth convex function using proximal-gradient methods, where an error is present in the calculation of the gradient of the smooth term or in the proximity operator with respect to the second term. We show that the basic proximal-gradient method, the basic proximal-gradient method with a strong convexity assumption, and the accelerated proximal-gradient method achieve the same convergence rates as in the error-free case, provided the errors decrease at an appropriate rate. Our experimental results on a structured sparsity problem indicate that sequences of errors with these appealing theoretical properties can lead to practical performance improvements.

We study the empirical strategies that humans follow as they teach a target concept with a simple 1D threshold to a robot. Previous studies of computational teaching, particularly the teaching dimension model and the curriculum learning principle, offer contradictory predictions on what optimal strategy the teacher should follow in this teaching task. We show through behavioral studies that humans employ three distinct teaching strategies, one of which is consistent with the curriculum learning principle, and propose a novel theoretical framework as a potential explanation for this strategy. This framework, which assumes a teaching goal of minimizing the learner's expected generalization error at each iteration, extends the standard teaching dimension model and offers a theoretical justification for curriculum learning.