As an automatic method of determining model complexity using the training data alone, Bayesian linear regression provides us a principled way to select hyperparameters. But one often needs approximation inference if distribution assumption is beyond Gaussian distribution. In this paper, we propose a Bayesian linear regression model with Student-t assumptions (BLRS), which can be inferred exactly. In this framework, both conjugate prior and expectation maximization (EM) algorithm are generalized. Meanwhile, we prove that the maximum likelihood solution is equivalent to the standard Bayesian linear regression with Gaussian assumptions (BLRG). The $q$-EM algorithm for BLRS is nearly identical to the EM algorithm for BLRG. It is showed that $q$-EM for BLRS can converge faster than EM for BLRG for the task of predicting online news popularity.

We study the minimization of a convex function $f(X)$ over the set of $n\times n$ positive semi-definite matrices, but when the problem is recast as $\min_U g(U) := f(UU^\top)$, with $U \in \mathbb{R}^{n \times r}$ and $r \leq n$. We study the performance of gradient descent on $g$---which we refer to as Factored Gradient Descent (FGD)---under standard assumptions on the original function $f$. We provide a rule for selecting the step size and, with this choice, show that the local convergence rate of FGD mirrors that of standard gradient descent on the original $f$: i.e., after $k$ steps, the error is $O(1/k)$ for smooth $f$, and exponentially small in $k$ when $f$ is (restricted) strongly convex. In addition, we provide a procedure to initialize FGD for (restricted) strongly convex objectives and when one only has access to $f$ via a first-order oracle; for several problem instances, such proper initialization leads to global convergence guarantees. FGD and similar procedures are widely used in practice for problems that can be posed as matrix factorization. To the best of our knowledge, this is the first paper to provide precise convergence rate guarantees for general convex functions under standard convex assumptions.

Gaussian processes allow for flexible specification of prior assumptions of unknown dynamics in state space models. We present a procedure for efficient Bayesian learning in Gaussian process state space models, where the representation is formed by projecting the problem onto a set of approximate eigenfunctions derived from the prior covariance structure. Learning under this family of models can be conducted using a carefully crafted particle MCMC algorithm. This scheme is computationally efficient and yet allows for a fully Bayesian treatment of the problem. Compared to conventional system identification tools or existing learning methods, we show competitive performance and reliable quantification of uncertainties in the model.

This paper addresses the problem of localizing audio sources using binaural measurements. We propose a supervised formulation that simultaneously localizes multiple sources at different locations. The approach is intrinsically efficient because, contrary to prior work, it relies neither on source separation, nor on monaural segregation. The method starts with a training stage that establishes a locally-linear Gaussian regression model between the directional coordinates of all the sources and the auditory features extracted from binaural measurements. While fixed-length wide-spectrum sounds (white noise) are used for training to reliably estimate the model parameters, we show that the testing (localization) can be extended to variable-length sparse-spectrum sounds (such as speech), thus enabling a wide range of realistic applications. Indeed, we demonstrate that the method can be used for audio-visual fusion, namely to map speech signals onto images and hence to spatially align the audio and visual modalities, thus enabling to discriminate between speaking and non-speaking faces. We release a novel corpus of real-room recordings that allow quantitative evaluation of the co-localization method in the presence of one or two sound sources. Experiments demonstrate increased accuracy and speed relative to several state-of-the-art methods.

In this paper we consider the problem of data clustering under the union of subspaces (UOS) model [1], [2], when each data vector is sampled in an element-wise manner. This is referred to as the case of missing data. In other words we are looking to harvest a union of subspaces structure from the data, when the data is missing. Such a problem has been recently considered in a number of papers [3], [4], [5], [6]. This setting has implications to data completion under the union of subspaces model in contrast to the single subspace model that has been prevalent in the matrix completion literature. In contrast to statistical analysis in [3], [4], [5], this paper uses a variant of the sparse subspace clustering (SSC) algorithm [2] to give sufficient deterministic conditions for accurate subspace clustering under missing data. In contrast to [6], which does not provide any specific conditions for success of SSC under missing data, in this paper we provide implications of the deterministic conditions for several specific cases of sampling. Further through extensive simulations we demonstrate for the first time that accurate clustering under missing data does not imply accurate subspace clustering and completion thereby indicating the natural order of hardness of these problems under missing data.

When we go grocery shopping, we often have a standard list of things to buy. Each shopper has a distinctive list, depending on one's needs and preferences. A housewife might buy healthy ingredients for a family dinner, while a bachelor might buy beer and chips. Understanding these buying patterns can help to increase sales in several ways. While we may know that certain items are frequently bought together, the question is, how do we uncover these associations? Besides increasing sales profits, association rules can also be used in other fields.

We humans may still be licking our wounds following AI's victory at the ancient game of Go, but it turns out we still have something to be proud of: We're doing a lot better than machines are at solving some of the key problems of quantum computing. Quantum mechanics are notoriously mind-bending because so-called "qubits" -- the atomic-scale building blocks of quantum computers -- can inhabit more than one physical state at once. That's known as superposition, and it's what gives the prospect of quantum computers their exciting potential. It's just potential at this point, however, because there are still many, many challenges to be solved before we can create a working quantum computer. That's where gaming comes in.

Vast swarms of miniature robots are coming -- and they might be the answer to scrubbing our waters clean of lead. "Microbots" smaller than the width of a human hair could be highly effective and cost-efficient tools for removing lead and other contaminants from industrial wastewater, according to a new study published in the journal Nano Letters last month. In the space of a single hour, the study showed, self-propelled microbots could remove up to 95 percent of lead from water. Lead is commonly found in wastewater from mines or factories that make batteries and electronic devices, and can pose a serious risk to public health, as the water crisis in Flint, Michigan demonstrates. Heavy metal pollution can cost big cities billions of dollars a year, said Samuel Sánchez, co-author of the study and a research group leader at the Max Planck Institute for Intelligent Systems in Germany.

There are so many deep learning libraries to choose from. Which are the good professional libraries that are worth learning and which are someones side project and should be avoided. It is hard to tell the difference. In this post you will discover the top deep learning libraries that you should consider learning and using in your own deep learning project. Popular Deep Learning Libraries Photo by Nikki, some rights reserved.

Dreyer says training a machine algorithm is similar to teaching a child, in that it requires considerable repetition. For example, by training an artificial neural network with thousands of images of correctly and incorrectly placed central lines, these algorithms can "learn" the difference between the two. While the human brain is optimized for the interpretation of visual data, these cognitive computations can be optimized to quantify nearly any data while eliminating the judgment and fatigue of humans.