We address the problem of estimating the ratio of two probability density functions (a.k.a.~the importance). The importance values can be used for various succeeding tasks such as non-stationarity adaptation or outlier detection. In this paper, we propose a new importance estimation method that has a closed-form solution; the leave-one-out cross-validation score can also be computed analytically. Therefore, the proposed method is computationally very efficient and numerically stable. We also elucidate theoretical properties of the proposed method such as the convergence rate and approximation error bound. Numerical experiments show that the proposed method is comparable to the best existing method in accuracy, while it is computationally more efficient than competing approaches.

This paper examines the generalization properties of online convex programming algorithms when the loss function is Lipschitz and strongly convex. Our main result is a sharp bound, that holds with high probability, on the excess risk of the output of an online algorithm in terms of the average regret. This allows one to use recent algorithms with logarithmic cumulative regret guarantees to achieve fast convergence rates for the excess risk with high probability. As a corollary, we characterize the convergence rate of PEGASOS (with high probability), a recently proposed method for solving the SVM optimization problem.

We provide sharp bounds for Rademacher and Gaussian complexities of (constrained) linear classes. These bounds make short work of providing a number of corollaries including: risk bounds for linear prediction (including settings where the weight vectors are constrained by either $L_2$ or $L_1$ constraints), margin bounds (including both $L_2$ and $L_1$ margins, along with more general notions based on relative entropy), a proof of the PAC-Bayes theorem, and $L_2$ covering numbers (with $L_p$ norm constraints and relative entropy constraints). In addition to providing a unified analysis, the results herein provide some of the sharpest risk and margin bounds (improving upon a number of previous results). Interestingly, our results show that the uniform convergence rates of empirical risk minimization algorithms tightly match the regret bounds of online learning algorithms for linear prediction (up to a constant factor of 2).

In most cognitive and motor tasks, speed-accuracy tradeoffs are observed: Individuals can respond slowly and accurately, or quickly yet be prone to errors. Control mechanisms governing the initiation of behavioral responses are sensitive not only to task instructions and the stimulus being processed, but also to the recent stimulus history. When stimuli can be characterized on an easy-hard dimension (e.g., word frequency in a naming task), items preceded by easy trials are responded to more quickly, and with more errors, than items preceded by hard trials. We propose a rationally motivated mathematical model of this sequential adaptation of control, based on a diffusion model of the decision process in which difficulty corresponds to the drift rate for the correct response. The model assumes that responding is based on the posterior distribution over which response is correct, conditioned on the accumulated evidence. We derive this posterior as a function of the drift rate, and show that higher estimates of the drift rate lead to (normatively) faster responding. Trial-by-trial tracking of difficulty thus leads to sequential effects in speed and accuracy. Simulations show the model explains a variety of phenomena in human speeded decision making. We argue this passive statistical mechanism provides a more elegant and parsimonious account than extant theories based on elaborate control structures.

We present a multi-label multiple kernel learning (MKL) formulation, in which the data are embedded into a low-dimensional space directed by the instance-label correlations encoded into a hypergraph. We formulate the problem in the kernel-induced feature space and propose to learn the kernel matrix as a linear combination of a given collection of kernel matrices in the MKL framework. The proposed learning formulation leads to a non-smooth min-max problem, and it can be cast into a semi-infinite linear program (SILP). We further propose an approximate formulation with a guaranteed error bound which involves an unconstrained and convex optimization problem. In addition, we show that the objective function of the approximate formulation is continuously differentiable with Lipschitz gradient, and hence existing methods can be employed to compute the optimal solution efficiently. We apply the proposed formulation to the automated annotation of Drosophila gene expression pattern images, and promising results have been reported in comparison with representative algorithms.

We present an approach to low-level vision that combines two main ideas: the use of convolutional networks as an image processing architecture and an unsupervised learning procedure that synthesizes training samples from specific noise models. We demonstrate this approach on the challenging problem of natural image denoising. Using a test set with a hundred natural images, we find that convolutional networks provide comparable and in some cases superior performance to state of the art wavelet and Markov random field (MRF) methods. Moreover, we find that a convolutional network offers similar performance in the blind denoising setting as compared to other techniques in the non-blind setting. We also show how convolutional networks are mathematically related to MRF approaches by presenting a mean field theory for an MRF specially designed for image denoising. Although these approaches are related, convolutional networks avoid computational difficulties in MRF approaches that arise from probabilistic learning and inference. This makes it possible to learn image processing architectures that have a high degree of representational power (we train models with over 15,000 parameters), but whose computational expense is significantly less than that associated with inference in MRF approaches with even hundreds of parameters.

Metric learning algorithms can provide useful distance functions for a variety of domains, and recent work has shown good accuracy for problems where the learner can access all distance constraints at once. However, in many real applications, constraints are only available incrementally, thus necessitating methods that can perform online updates to the learned metric. Existing online algorithms offer bounds on worst-case performance, but typically do not perform well in practice as compared to their offline counterparts. We present a new online metric learning algorithm that updates a learned Mahalanobis metric based on LogDet regularization and gradient descent. We prove theoretical worst-case performance bounds, and empirically compare the proposed method against existing online metric learning algorithms. To further boost the practicality of our approach, we develop an online locality-sensitive hashing scheme which leads to efficient updates for approximate similarity search data structures. We demonstrate our algorithm on multiple datasets and show that it outperforms relevant baselines.

In multi-task learning several related tasks are considered simultaneously, with the hope that by an appropriate sharing of information across tasks, each task may benefit from the others. In the context of learning linear functions for supervised classification or regression, this can be achieved by including a priori information about the weight vectors associated with the tasks, and how they are expected to be related to each other. In this paper, we assume that tasks are clustered into groups, which are unknown beforehand, and that tasks within a group have similar weight vectors. We design a new spectral norm that encodes this a priori assumption, without the prior knowledge of the partition of tasks into groups, resulting in a new convex optimization formulation for multi-task learning. We show in simulations on synthetic examples and on the iedb MHC-I binding dataset, that our approach outperforms well-known convex methods for multi-task learning, as well as related non convex methods dedicated to the same problem.

The visual and auditory map alignment in the Superior Colliculus (SC) of barn owl is important for its accurate localization for prey behavior. Prism learning or Blindness may interfere this alignment and cause loss of the capability of accurate prey. However, juvenile barn owl could recover its sensory map alignment by shifting its auditory map. The adaptation of this map alignment is believed based on activity dependent axon developing in Inferior Colliculus (IC). A model is built to explore this mechanism. In this model, axon growing process is instructed by an inhibitory network in SC while the strength of the inhibition adjusted by Spike Timing Dependent Plasticity (STDP). We test and analyze this mechanism by application of the neural structures involved in spatial localization in a robotic system.

Spectral clustering is useful for a wide-ranging set of applications in areas such as biological data analysis, image processing and data mining. However, the computational and/or communication resources required by the method in processing large-scale data sets are often prohibitively high, and practitioners are often required to perturb the original data in various ways (quantization, downsampling, etc) before invoking a spectral algorithm. In this paper, we use stochastic perturbation theory to study the effects of data perturbation on the performance of spectral clustering. We show that the error under perturbation of spectral clustering is closely related to the perturbation of the eigenvectors of the Laplacian matrix. From this result we derive approximate upper bounds on the clustering error. We show that this bound is tight empirically across a wide range of problems, suggesting that it can be used in practical settings to determine the amount of data reduction allowed in order to meet a specification of permitted loss in clustering performance.