Optimal Trade-offs in Multi-Processor Approximate Message Passing

We consider large-scale linear inverse problems in Bayesian settings. We follow a recent line of work that applies the approximate message passing (AMP) framework to multi-processor (MP) computational systems, where each processor node stores and processes a subset of rows of the measurement matrix along with corresponding measurements. In each MP-AMP iteration, nodes of the MP system and its fusion center exchange lossily compressed messages pertaining to their estimates of the input. In this setup, we derive the optimal per-iteration coding rates using dynamic programming. We analyze the excess mean squared error (EMSE) beyond the minimum mean squared error (MMSE), and prove that, in the limit of low EMSE, the optimal coding rates increase approximately linearly per iteration. Additionally, we obtain that the combined cost of computation and communication scales with the desired estimation quality according to $O(\log^2(1/\text{EMSE}))$. Finally, we study trade-offs between the physical costs of the estimation process including computation time, communication loads, and the estimation quality as a multi-objective optimization problem, and characterize the properties of the Pareto optimal surfaces.Comment: 14 pages, 8 figure

LQR based improved discrete PID controller design via optimum selection of weighting matrices using fractional order integral performance index

The continuous and discrete time Linear Quadratic Regulator (LQR) theory has been used in this paper for the design of optimal analog and discrete PID controllers respectively. The PID controller gains are formulated as the optimal state-feedback gains, corresponding to the standard quadratic cost function involving the state variables and the controller effort. A real coded Genetic Algorithm (GA) has been used next to optimally find out the weighting matrices, associated with the respective optimal state-feedback regulator design while minimizing another time domain integral performance index, comprising of a weighted sum of Integral of Time multiplied Squared Error (ITSE) and the controller effort. The proposed methodology is extended for a new kind of fractional order (FO) integral performance indices. The impact of fractional order (as any arbitrary real order) cost function on the LQR tuned PID control loops is highlighted in the present work, along with the achievable cost of control. Guidelines for the choice of integral order of the performance index are given depending on the characteristics of the process, to be controlled.Comment: 22 pages, 12 figure

Non-linear dimensionality reduction: Riemannian metric estimation and the problem of geometric discovery

In recent years, manifold learning has become increasingly popular as a tool for performing non-linear dimensionality reduction. This has led to the development of numerous algorithms of varying degrees of complexity that aim to recover man ifold geometry using either local or global features of the data. Building on the Laplacian Eigenmap and Diffusionmaps framework, we propose a new paradigm that offers a guarantee, under reasonable assumptions, that any manifo ld learning algorithm will preserve the geometry of a data set. Our approach is based on augmenting the output of embedding algorithms with geometric informatio n embodied in the Riemannian metric of the manifold. We provide an algorithm for estimating the Riemannian metric from data and demonstrate possible application s of our approach in a variety of examples.Comment: 32 page

Persistent-Homology-based Machine Learning and its Applications -- A Survey

A suitable feature representation that can both preserve the data intrinsic information and reduce data complexity and dimensionality is key to the performance of machine learning models. Deeply rooted in algebraic topology, persistent homology (PH) provides a delicate balance between data simplification and intrinsic structure characterization, and has been applied to various areas successfully. However, the combination of PH and machine learning has been hindered greatly by three challenges, namely topological representation of data, PH-based distance measurements or metrics, and PH-based feature representation. With the development of topological data analysis, progresses have been made on all these three problems, but widely scattered in different literatures. In this paper, we provide a systematical review of PH and PH-based supervised and unsupervised models from a computational perspective. Our emphasizes are the recent development of mathematical models and tools, including PH softwares and PH-based functions, feature representations, kernels, and similarity models. Essentially, this paper can work as a roadmap for the practical application of PH-based machine learning tools. Further, we consider different topological feature representations in different machine learning models, and investigate their impacts on the protein secondary structure classification.Comment: 42 pages; 6 figures; 9 table

The Bayesian update: variational formulations and gradient flows

The Bayesian update can be viewed as a variational problem by characterizing the posterior as the minimizer of a functional. The variational viewpoint is far from new and is at the heart of popular methods for posterior approximation. However, some of its consequences seem largely unexplored. We focus on the following one: defining the posterior as the minimizer of a functional gives a natural path towards the posterior by moving in the direction of steepest descent of the functional. This idea is made precise through the theory of gradient flows, allowing to bring new tools to the study of Bayesian models and algorithms. Since the posterior may be characterized as the minimizer of different functionals, several variational formulations may be considered. We study three of them and their three associated gradient flows. We show that, in all cases, the rate of convergence of the flows to the posterior can be bounded by the geodesic convexity of the functional to be minimized. Each gradient flow naturally suggests a nonlinear diffusion with the posterior as invariant distribution. These diffusions may be discretized to build proposals for Markov chain Monte Carlo (MCMC) algorithms. By construction, the diffusions are guaranteed to satisfy a certain optimality condition, and rates of convergence are given by the convexity of the functionals. We use this observation to propose a criterion for the choice of metric in Riemannian MCMC methods

Orientation-boosted Voxel Nets for 3D Object Recognition

Recent work has shown good recognition results in 3D object recognition using 3D convolutional networks. In this paper, we show that the object orientation plays an important role in 3D recognition. More specifically, we argue that objects induce different features in the network under rotation. Thus, we approach the category-level classification task as a multi-task problem, in which the network is trained to predict the pose of the object in addition to the class label as a parallel task. We show that this yields significant improvements in the classification results. We test our suggested architecture on several datasets representing various 3D data sources: LiDAR data, CAD models, and RGB-D images. We report state-of-the-art results on classification as well as significant improvements in precision and speed over the baseline on 3D detection.Comment: BMVC'17 version. Added some experiments + auto-alignment of Modelnet4

Latent Skill Embedding for Personalized Lesson Sequence Recommendation

Students in online courses generate large amounts of data that can be used to personalize the learning process and improve quality of education. In this paper, we present the Latent Skill Embedding (LSE), a probabilistic model of students and educational content that can be used to recommend personalized sequences of lessons with the goal of helping students prepare for specific assessments. Akin to collaborative filtering for recommender systems, the algorithm does not require students or content to be described by features, but it learns a representation using access traces. We formulate this problem as a regularized maximum-likelihood embedding of students, lessons, and assessments from historical student-content interactions. An empirical evaluation on large-scale data from Knewton, an adaptive learning technology company, shows that this approach predicts assessment results competitively with benchmark models and is able to discriminate between lesson sequences that lead to mastery and failure.Comment: Under review by the ACM SIGKDD Conference on Knowledge Discovery and Data Minin

TATi-Thermodynamic Analytics ToolkIt: TensorFlow-based software for posterior sampling in machine learning applications

With the advent of GPU-assisted hardware and maturing high-efficiency software platforms such as TensorFlow and PyTorch, Bayesian posterior sampling for neural networks becomes plausible. In this article we discuss Bayesian parametrization in machine learning based on Markov Chain Monte Carlo methods, specifically discretized stochastic differential equations such as Langevin dynamics and extended system methods in which an ensemble of walkers is employed to enhance sampling. We provide a glimpse of the potential of the sampling-intensive approach by studying (and visualizing) the loss landscape of a neural network applied to the MNIST data set. Moreover, we investigate how the sampling efficiency itself can be significantly enhanced through an ensemble quasi-Newton preconditioning method. This article accompanies the release of a new TensorFlow software package, the Thermodynamic Analytics ToolkIt, which is used in the computational experiments.Comment: 25 pages: textual improvements with results unchanged, sections on TATi architecture and software performance removed for size constraints, extended EQN parts, added MNIST nonlinear perceptron exampl

Deep Learning for Passive Synthetic Aperture Radar

We introduce a deep learning (DL) framework for inverse problems in imaging, and demonstrate the advantages and applicability of this approach in passive synthetic aperture radar (SAR) image reconstruction. We interpret image recon- struction as a machine learning task and utilize deep networks as forward and inverse solvers for imaging. Specifically, we design a recurrent neural network (RNN) architecture as an inverse solver based on the iterations of proximal gradient descent optimization methods. We further adapt the RNN architecture to image reconstruction problems by transforming the network into a recurrent auto-encoder, thereby allowing for unsupervised training. Our DL based inverse solver is particularly suitable for a class of image formation problems in which the forward model is only partially known. The ability to learn forward models and hyper parameters combined with unsupervised training approach establish our recurrent auto-encoder suitable for real world applications. We demonstrate the performance of our method in passive SAR image reconstruction. In this regime a source of opportunity, with unknown location and transmitted waveform, is used to illuminate a scene of interest. We investigate recurrent auto- encoder architecture based on the 1 and 0 constrained least- squares problem. We present a projected stochastic gradient descent based training scheme which incorporates constraints of the unknown model parameters. We demonstrate through extensive numerical simulations that our DL based approach out performs conventional sparse coding methods in terms of computation and reconstructed image quality, specifically, when no information about the transmitter is available.Comment: Submitted to IEEE Journal of Selected Topics in Signal Processin

A Scala Prototype to Generate Multigrid Solver Implementations for Different Problems and Target Multi-Core Platforms

Many problems in computational science and engineering involve partial differential equations and thus require the numerical solution of large, sparse (non)linear systems of equations. Multigrid is known to be one of the most efficient methods for this purpose. However, the concrete multigrid algorithm and its implementation highly depend on the underlying problem and hardware. Therefore, changes in the code or many different variants are necessary to cover all relevant cases. In this article we provide a prototype implementation in Scala for a framework that allows abstract descriptions of PDEs, their discretization, and their numerical solution via multigrid algorithms. From these, one is able to generate data structures and implementations of multigrid components required to solve elliptic PDEs on structured grids. Two different test problems showcase our proposed automatic generation of multigrid solvers for both CPU and GPU target platforms