The human auditory system is able to distinguish the vocal source of thousands of speakers, yet not much is known about what features the auditory system uses to do this. Fourier Transforms are capable of capturing the pitch and harmonic structure of the speaker but this alone proves insufficient at identifying speakers uniquely. The remaining structure, often referred to as timbre, is critical to identifying speakers but we understood little about it. In this paper we use recent advances in neural networks in order to manipulate the voice of one speaker into another by transforming not only the pitch of the speaker, but the timbre. We review generative models built with neural networks as well as architectures for creating neural networks that learn analogies. Our preliminary results converting voices from one speaker to another are encouraging.

Positive-definite kernel functions are fundamental elements of kernel methods and Gaussian processes. A well-known construction of such functions comes from Bochner's characterization, which connects a positive-definite function with a probability distribution. Another construction, which appears to have attracted less attention, is Polya's criterion that characterizes a subset of these functions. In this paper, we study the latter characterization and derive a number of novel kernels little known previously. In the context of large-scale kernel machines, Rahimi and Recht (2007) proposed a random feature map (random Fourier) that approximates a kernel function, through independent sampling of the probability distribution in Bochner's characterization. The authors also suggested another feature map (random binning), which, although not explicitly stated, comes from Polya's characterization. We show that with the same number of random samples, the random binning map results in an Euclidean inner product closer to the kernel than does the random Fourier map. The superiority of the random binning map is confirmed empirically through regressions and classifications in the reproducing kernel Hilbert space.

Supervised dimensionality reduction has emerged as an important theme in the last decade. Despite the plethora of models and formulations, there is a lack of a simple model which aims to project the set of patterns into a space defined by the classes (or categories). To this end, we set up a model in which each class is represented as a 1D subspace of the vector space formed by the features. Assuming the set of classes does not exceed the cardinality of the features, the model results in multi-class supervised learning in which the features of each class are projected into the class subspace. Class discrimination is automatically guaranteed via the imposition of orthogonality of the 1D class sub-spaces. The resulting optimization problem - formulated as the minimization of a sum of quadratic functions on a Stiefel manifold - while being non-convex (due to the constraints), nevertheless has a structure for which we can identify when we have reached a global minimum. After formulating a version with standard inner products, we extend the formulation to reproducing kernel Hilbert spaces in a straightforward manner. The optimization approach also extends in a similar fashion to the kernel version. Results and comparisons with the multi-class Fisher linear (and kernel) discriminants and principal component analysis (linear and kernel) showcase the relative merits of this approach to dimensionality reduction.

Signal subspace methods (SSM) are efficient techniques to reduce dimensionality of data and to filter out noise [1]. The fundamental idea under SSM is to project the data on a basis made of two subspaces, one mostly containing the signal and the other the noise. The two subspaces are separated by a thresholding criterion associated with some measures of information. The two most popular methods of signal subspace decomposition are wavelet shrinkage [2] and Principal Component Analysis (PCA) [3]. Both techniques have proved to be quite efficient. However, wavelet decomposition depending on signal statistics is not equally adapted to different data, and requires some knowledge on prior distributions or parameters of signals to efficiently choose the thresholds for shrinkage. A significant advantage of the PCA is its adaptability to data. The separation criterion is based on energy which may be seen as a limitation in some cases as illustrated in the next section. In recent years, sparse coding has attracted significant interest in the field of signal denoising [4].

There are now many publicly available Gaussian process libraries ranging in scale from personal projects to major community tools. We will therefore only consider a relevant subset of the existing libraries. The influential GPML toolbox (Rasmussen and Nickisch, 2010) uses MATLAB. It has been widely forked. A key reference for our particular contribution is the GPy library (GPy, since 2012), which is written primarily using Python and Numeric Python (NumPy). GPy has an intuitive object-oriented interface. Another relevant Gaussian process library is GPstuff (Vanhatalo et al., 2013) which is also a MATLAB library.

The stochastic gradient descent (SGD) algorithm has been widely used in statistical estimation for large-scale data due to its computational and memory efficiency. While most existing work focuses on the convergence of the objective function or the error of the obtained solution, we investigate the problem of statistical inference of the true model parameters based on SGD. To this end, we propose two consistent estimators of the asymptotic covariance of the average iterate from SGD: (1) an intuitive plug-in estimator and (2) a computationally more efficient batch-means estimator, which only uses the iterates from SGD. As the SGD process forms a time-inhomogeneous Markov chain, our batch-means estimator with carefully chosen increasing batch sizes generalizes the classical batch-means estimator designed for time-homogenous Markov chains. The proposed batch-means estimator is of independent interest, which can be potentially used for estimating the covariance of other time-inhomogeneous Markov chains. Both proposed estimators allow us to construct asymptotically exact confidence intervals and hypothesis tests. We further discuss an extension to conducting inference based on SGD for high-dimensional linear regression. Using a variant of the SGD algorithm, we construct a debiased estimator of each regression coefficient that is asymptotically normal. This gives a one-pass algorithm for computing both the sparse regression coefficient estimator and confidence intervals, which is computationally attractive and applicable to online data.

We consider the problem of transfer learning in an online setting. Different tasks are presented sequentially and processed by a within-task algorithm. We propose a lifelong learning strategy which refines the underlying data representation used by the within-task algorithm, thereby transferring information from one task to the next. We show that when the within-task algorithm comes with some regret bound, our strategy inherits this good property. Our bounds are in expectation for a general loss function, and uniform for a convex loss. We discuss applications to dictionary learning and finite set of predictors. In the latter case, we improve previous $O(1/\sqrt{m})$ bounds to $O(1/m)$ where $m$ is the per task sample size.

In this paper, we take a new look at the possibilistic c-means (PCM) and adaptive PCM (APCM) clustering algorithms from the perspective of uncertainty. This new perspective offers us insights into the clustering process, and also provides us greater degree of flexibility. We analyze the clustering behavior of PCM-based algorithms and introduce parameters $\sigma_v$ and $\alpha$ to characterize uncertainty of estimated bandwidth and noise level of the dataset respectively. Then uncertainty (fuzziness) of membership values caused by uncertainty of the estimated bandwidth parameter is modeled by a conditional fuzzy set, which is a new formulation of the type-2 fuzzy set. Experiments show that parameters $\sigma_v$ and $\alpha$ make the clustering process more easy to control, and main features of PCM and APCM are unified in this new clustering framework (UPCM). More specifically, UPCM reduces to PCM when we set a small $\alpha$ or a large $\sigma_v$, and UPCM reduces to APCM when clusters are confined in their physical clusters and possible cluster elimination are ensured. Finally we present further researches of this paper.

In this paper, we discuss structure learning of causal networks from multiple data sets obtained by external intervention experiments where we do not know what variables are manipulated. For example, the conditions in these experiments are changed by changing temperature or using drugs, but we do not know what target variables are manipulated by the external interventions. From such data sets, the structure learning becomes more difficult. For this case, we first discuss the identifiability of causal structures. Next we present a graph-merging method for learning causal networks for the case that the sample sizes are large for these interventions. Then for the case that the sample sizes of these interventions are relatively small, we propose a data-pooling method for learning causal networks in which we pool all data sets of these interventions together for the learning. Further we propose a re-sampling approach to evaluate the edges of the causal network learned by the data-pooling method. Finally we illustrate the proposed learning methods by simulations.

The problem of forecasting conditional probabilities of the next event given the past is considered in a general probabilistic setting. Given an arbitrary (large, uncountable) set C of predictors, we would like to construct a single predictor that performs asymptotically as well as the best predictor in C, on any data. Here we show that there are sets C for which such predictors exist, but none of them is a Bayesian predictor with a prior concentrated on C. In other words, there is a predictor with sublinear regret, but every Bayesian predictor must have a linear regret. This negative finding is in sharp contrast with previous results that establish the opposite for the case when one of the predictors in $C$ achieves asymptotically vanishing error. In such a case, if there is a predictor that achieves asymptotically vanishing error for any measure in C, then there is a Bayesian predictor that also has this property, and whose prior is concentrated on (a countable subset of) C.