Traditional statistical wavelet analysis that carries out modeling and inference based on wavelet coefficients under a given, predetermined wavelet transform can quickly lose efficiency in multivariate problems, because such wavelet transforms, which are typically symmetric with respect to the dimensions, cannot adaptively exploit the energy distribution in a problem-specific manner. We introduce a principled probabilistic framework for incorporating such adaptivity---by (i) representing multivariate functions using one-dimensional (1D) wavelet transforms applied to a permuted version of the original function, and (ii) placing a prior on the corresponding permutation, thereby forming a mixture of permuted 1D wavelet transforms. Such a representation can achieve substantially better energy concentration in the wavelet coefficients. In particular, when combined with the Haar basis, we show that exact Bayesian inference under the model can be achieved analytically through a recursive message passing algorithm with a computational complexity that scales linearly with sample size. In addition, we propose a sequential Monte Carlo (SMC) inference algorithm for other wavelet bases using the exact Haar solution as the proposal. We demonstrate that with this framework even simple 1D Haar wavelets can achieve excellent performance in both 2D and 3D image reconstruction via numerical experiments, outperforming state-of-the-art multidimensional wavelet-based methods especially in low signal-to-noise ratio settings, at a fraction of the computational cost.

We consider the problem of inference in a causal generative model where the set of available observations differs between data instances. We show how combining samples drawn from the graphical model with an appropriate masking function makes it possible to train a single neural network to approximate all the corresponding conditional marginal distributions and thus amortize the cost of inference. We further demonstrate that the efficiency of importance sampling may be improved by basing proposals on the output of the neural network. We also outline how the same network can be used to generate samples from an approximate joint posterior via a chain decomposition of the graph.

This paper proposes a method for multi-class classification problems, where the number of classes $K$ is large. The method, referred to as {\em Candidates v.s. Noises Estimation} (CANE), selects a small subset of candidate classes and samples the remaining classes. We show that CANE is always consistent and computationally efficient. Moreover, the resulting estimator has low statistical variance approaching that of the maximum likelihood estimator, when the observed label belongs to the selected candidates with high probability. In practice, we use a tree structure with leaves as classes to promote fast beam search for candidate selection. We also apply the CANE method to estimate word probabilities in neural language models. Experiments show that CANE achieves better prediction accuracy over the Noise-Contrastive Estimation (NCE), its variants and a number of the state-of-the-art tree classifiers, while it gains significant speedup compared to the standard $\mathcal{O}(K)$ methods.

This paper presents a novel variational inference framework for deriving a family of Bayesian sparse Gaussian process regression (SGPR) models whose approximations are variationally optimal with respect to the full-rank GPR model enriched with various corresponding correlation structures of the observation noises. Our variational Bayesian SGPR (VBSGPR) models jointly treat both the distributions of the inducing variables and hyperparameters as variational parameters, which enables the decomposability of the variational lower bound that in turn can be exploited for stochastic optimization. Such a stochastic optimization involves iteratively following the stochastic gradient of the variational lower bound to improve its estimates of the optimal variational distributions of the inducing variables and hyperparameters (and hence the predictive distribution) of our VBSGPR models and is guaranteed to achieve asymptotic convergence to them. We show that the stochastic gradient is an unbiased estimator of the exact gradient and can be computed in constant time per iteration, hence achieving scalability to big data. We empirically evaluate the performance of our proposed framework on two real-world, massive datasets.

Lifelong learning is the problem of learning multiple consecutive tasks in a sequential manner where knowledge gained from previous tasks is retained and used for future learning. It is essential towards the development of intelligent machines that can adapt to their surroundings. In this work we focus on a lifelong learning approach to generative modeling where we continuously incorporate newly observed streaming distributions into our learnt model. We do so through a student-teacher architecture which allows us to learn and preserve all the distributions seen so far without the need to retain the past data nor the past models. Through the introduction of a novel cross-model regularizer, the student model leverages the information learnt by the teacher, which acts as a summary of everything seen till now. The regularizer has the additional benefit of reducing the effect of catastrophic interference that appears when we learn over streaming data. We demonstrate its efficacy on streaming distributions as well as its ability to learn a common latent representation across a complex transfer learning scenario.

Meta learning of optimal classifier error rates allows an experimenter to empirically estimate the intrinsic ability of any estimator to discriminate between two populations, circumventing the difficult problem of estimating the optimal Bayes classifier. To this end we propose a weighted nearest neighbor (WNN) graph estimator for a tight bound on the Bayes classification error; the Henze-Penrose (HP) divergence. Similar to recently proposed HP estimators [berisha2016], the proposed estimator is non-parametric and does not require density estimation. However, unlike previous approaches the proposed estimator is rate-optimal, i.e., its mean squared estimation error (MSEE) decays to zero at the fastest possible rate of $O(1/M+1/N)$ where $M,N$ are the sample sizes of the respective populations. We illustrate the proposed WNN meta estimator for several simulated and real data sets.

This work constructs a hypothesis test for detecting whether an data-generating function $h: R^p \rightarrow R$ belongs to a specific reproducing kernel Hilbert space $\mathcal{H}_0$ , where the structure of $\mathcal{H}_0$ is only partially known. Utilizing the theory of reproducing kernels, we reduce this hypothesis to a simple one-sided score test for a scalar parameter, develop a testing procedure that is robust against the mis-specification of kernel functions, and also propose an ensemble-based estimator for the null model to guarantee test performance in small samples. To demonstrate the utility of the proposed method, we apply our test to the problem of detecting nonlinear interaction between groups of continuous features. We evaluate the finite-sample performance of our test under different data-generating functions and estimation strategies for the null model. Our results reveal interesting connections between notions in machine learning (model underfit/overfit) and those in statistical inference (i.e. Type I error/power of hypothesis test), and also highlight unexpected consequences of common model estimating strategies (e.g. estimating kernel hyperparameters using maximum likelihood estimation) on model inference.

Reward augmented maximum likelihood (RAML), a simple and effective learning framework to directly optimize towards the reward function in structured prediction tasks, has led to a number of impressive empirical successes. RAML incorporates task-specific reward by performing maximum-likelihood updates on candidate outputs sampled according to an exponentiated payoff distribution, which gives higher probabilities to candidates that are close to the reference output. While RAML is notable for its simplicity, efficiency, and its impressive empirical successes, the theoretical properties of RAML, especially the behavior of the exponentiated payoff distribution, has not been examined thoroughly. In this work, we introduce softmax Q-distribution estimation, a novel theoretical interpretation of RAML, which reveals the relation between RAML and Bayesian decision theory. The softmax Q-distribution can be regarded as a smooth approximation of the Bayes decision boundary, and the Bayes decision rule is achieved by decoding with this Q-distribution. We further show that RAML is equivalent to approximately estimating the softmax Q-distribution, with the temperature $\tau$ controlling approximation error. We perform two experiments, one on synthetic data of multi-class classification and one on real data of image captioning, to demonstrate the relationship between RAML and the proposed softmax Q-distribution estimation method, verifying our theoretical analysis. Additional experiments on three structured prediction tasks with rewards defined on sequential (named entity recognition), tree-based (dependency parsing) and irregular (machine translation) structures show notable improvements over maximum likelihood baselines.

We propose a probabilistic framework for domain adaptation that blends both generative and discriminative modeling in a principled way. Under this framework, generative and discriminative models correspond to specific choices of the prior over parameters. This provides us a very general way to interpolate between generative and discriminative extremes through different choices of priors. By maximizing both the marginal and the conditional log-likelihoods, models derived from this framework can use both labeled instances from the source domain as well as unlabeled instances from both source and target domains. Under this framework, we show that the popular reconstruction loss of autoencoder corresponds to an upper bound of the negative marginal log-likelihoods of unlabeled instances, where marginal distributions are given by proper kernel density estimations. This provides a way to interpret the empirical success of autoencoders in domain adaptation and semi-supervised learning. We instantiate our framework using neural networks, and build a concrete model, DAuto. Empirically, we demonstrate the effectiveness of DAuto on text, image and speech datasets, showing that it outperforms related competitors when domain adaptation is possible.

This work addresses the problem of segmentation in time series data with respect to a statistical parameter of interest in Bayesian models. It is common to assume that the parameters are distinct within each segment. As such, many Bayesian change point detection models do not exploit the segment parameter patterns, which can improve performance. This work proposes a Bayesian mean-shift change point detection algorithm that makes use of repetition in segment parameters, by introducing segment class labels that utilise a Dirichlet process prior. The performance of the proposed approach was assessed on both synthetic and real world data, highlighting the enhanced performance when using parameter labelling.