Functional Multi-Layer Perceptron: a Nonlinear Tool for Functional Data Analysis

In this paper, we study a natural extension of Multi-Layer Perceptrons (MLP) to functional inputs. We show that fundamental results for classical MLP can be extended to functional MLP. We obtain universal approximation results that show the expressive power of functional MLP is comparable to that of numerical MLP. We obtain consistency results which imply that the estimation of optimal parameters for functional MLP is statistically well defined. We finally show on simulated and real world data that the proposed model performs in a very satisfactory way.Comment: http://www.sciencedirect.com/science/journal/0893608

Interpretable and Steerable Sequence Learning via Prototypes

One of the major challenges in machine learning nowadays is to provide predictions with not only high accuracy but also user-friendly explanations. Although in recent years we have witnessed increasingly popular use of deep neural networks for sequence modeling, it is still challenging to explain the rationales behind the model outputs, which is essential for building trust and supporting the domain experts to validate, critique and refine the model. We propose ProSeNet, an interpretable and steerable deep sequence model with natural explanations derived from case-based reasoning. The prediction is obtained by comparing the inputs to a few prototypes, which are exemplar cases in the problem domain. For better interpretability, we define several criteria for constructing the prototypes, including simplicity, diversity, and sparsity and propose the learning objective and the optimization procedure. ProSeNet also provides a user-friendly approach to model steering: domain experts without any knowledge on the underlying model or parameters can easily incorporate their intuition and experience by manually refining the prototypes. We conduct experiments on a wide range of real-world applications, including predictive diagnostics for automobiles, ECG, and protein sequence classification and sentiment analysis on texts. The result shows that ProSeNet can achieve accuracy on par with state-of-the-art deep learning models. We also evaluate the interpretability of the results with concrete case studies. Finally, through user study on Amazon Mechanical Turk (MTurk), we demonstrate that the model selects high-quality prototypes which align well with human knowledge and can be interactively refined for better interpretability without loss of performance.Comment: Accepted as a full paper at KDD 2019 on May 8, 201

Anomalous Thermostat and Intraband Discrete Breathers

We investigate the dynamics of a macroscopic system which consists of an anharmonic subsystem embedded in an arbitrary harmonic lattice, including quenched disorder. Elimination of the harmonic degrees of freedom leads to a nonlinear Langevin equation for the anharmonic coordinates. For zero temperature, we prove that the support of the Fourier transform of the memory kernel and of the time averaged velocity-velocity correlations functions of the anharmonic system can not overlap. As a consequence, the asymptotic solutions can be constant, periodic,quasiperiodic or almost periodic, and possibly weakly chaotic. For a sinusoidal trajectory with frequency $\Omega$ we find that the energy $E_T$ transferred to the harmonic system up to time $T$ is proportional to $T^{\alpha}$. If $\Omega$ equals one of the phonon frequencies $\omega_\nu$, it is $\alpha=2$. We prove that there is a full measure set such that for $\Omega$ in this set it is $\alpha=0$, i.e. there is no energy dissipation. Under certain conditions there exists a zero measure set such that for $\Omega \in this set the dissipation rate is nonzero and may be subdissipative$(0 \leq \alpha < 1)$or superdissipative$(1 <\alpha \leq 2)$. Consequently, the harmonic bath does act as an anomalous thermostat. Intraband discrete breathers are such solutions which do not relax. We prove for arbitrary anharmonicity and small but finite coupling that intraband discrete breathers with frequency$\Omega$exist for all$\Omega$in a Cantor set$\mathcal{C}(k)$of finite Lebesgue measure. This is achieved by estimating the contribution of small denominators appearing in the memory kernel. For$\Omega\in\mathcal{C}(k)$the small denominators do not lead to divergencies such that this kernel is a smooth and bounded function in$t$.Comment: Physica D in prin

Quadratic Volume Preserving Maps

We study quadratic, volume preserving diffeomorphisms whose inverse is also quadratic. Such maps generalize the Henon area preserving map and the family of symplectic quadratic maps studied by Moser. In particular, we investigate a family of quadratic volume preserving maps in three space for which we find a normal form and study invariant sets. We also give an alternative proof of a theorem by Moser classifying quadratic symplectic maps.Comment: Ams LaTeX file with 4 figures (figure 2 is gif, the others are ps

On the flow map for 2D Euler equations with unbounded vorticity

In Part I, we construct a class of examples of initial velocities for which the unique solution to the Euler equations in the plane has an associated flow map that lies in no Holder space of positive exponent for any positive time. In Part II, we explore inverse problems that arise in attempting to construct an example of an initial velocity producing an arbitrarily poor modulus of continuity of the flow map.Comment: http://iopscience.iop.org/0951-7715/24/9/013/ for published versio

A Bishop-Phelps-Bollobas Type Theorem for uniform algegras

This paper is devoted to showing that Asplund operators with range in a uniform Banach algebra have the Bishop¿Phelps¿Bollobas property, i.e., they are approximated by norm attaining Asplund operators at the same time that a point where the approximated operator almost attains its norm is approximated by a point at which the approximating operator attains it. To prove this result we use the weak*-to-norm fragmentability of weak*-compact subsets of the dual of Asplund spaces and we need to observe a Urysohn type result producing peak complex-valued functions in uniform algebras that are small outside a given open set and whose image is inside a Stolz region.This research was partially supported by MEC and FEDER projects MTM2008-05396 and MTM2011-25377. The research of the second author was also partially supported by Generalitat Valenciana (GV/2010/036), and by Universidad Politecnica de Valencia (project PAID-06-09-2829).Cascales, B.; Guirao Sánchez, AJ.; Kadets, V. (2013). A Bishop-Phelps-Bollobas Type Theorem for uniform algegras. Advances in Mathematics. 240:370-382. https://doi.org/10.1016/j.aim.2013.03.005S37038224

Templates for Convex Cone Problems with Applications to Sparse Signal Recovery

This paper develops a general framework for solving a variety of convex cone problems that frequently arise in signal processing, machine learning, statistics, and other fields. The approach works as follows: first, determine a conic formulation of the problem; second, determine its dual; third, apply smoothing; and fourth, solve using an optimal first-order method. A merit of this approach is its flexibility: for example, all compressed sensing problems can be solved via this approach. These include models with objective functionals such as the total-variation norm, ||Wx||_1 where W is arbitrary, or a combination thereof. In addition, the paper also introduces a number of technical contributions such as a novel continuation scheme, a novel approach for controlling the step size, and some new results showing that the smooth and unsmoothed problems are sometimes formally equivalent. Combined with our framework, these lead to novel, stable and computationally efficient algorithms. For instance, our general implementation is competitive with state-of-the-art methods for solving intensively studied problems such as the LASSO. Further, numerical experiments show that one can solve the Dantzig selector problem, for which no efficient large-scale solvers exist, in a few hundred iterations. Finally, the paper is accompanied with a software release. This software is not a single, monolithic solver; rather, it is a suite of programs and routines designed to serve as building blocks for constructing complete algorithms.Comment: The TFOCS software is available at http://tfocs.stanford.edu This version has updated reference

A forgotten figure in Siouan and Caddoan linguistics: Samuel Stehman Haldeman.

The Siouan family comprises some twenty languages, historically spoken across a broad swath of the central North American plains and woodlands, as well as in parts of the southeastern United States. In spite of its geographical extent and diversity, and the size and importance of several Siouan-speaking tribes, this family has received relatively little attention in the linguistic literature and many of the individual Siouan languages are severely understudied. This volume aims to make work on Siouan languages more broadly available and to encourage deeper investigation of the myriad typological, theoretical, descriptive, and pedagogical issues they raise. The 17 chapters in this volume present a broad range of current Siouan research, focusing on various Siouan languages, from a variety of linguistic perspectives: historical-genetic, philological, applied, descriptive, formal/generative, and comparative/typological. The editors' preface summarizes characteristic features of the Siouan family, including head-final and "verb-centered" syntax, a complex system of verbal affixes including applicatives and subject-possessives, head-internal relative clauses, gendered speech markers, stop-systems including ejectives, and a preference for certain prosodic and phonotactic patterns. The volume is dedicated to the memory of Professor Robert L. Rankin, a towering figure in Siouan linguistics throughout his long career, who passed away in February of 2014

Advances in the study of Siouan languages and linguistics

The Siouan family comprises some twenty languages, historically spoken across a broad swath of the central North American plains and woodlands, as well as in parts of the southeastern United States. In spite of its geographical extent and diversity, and the size and importance of several Siouan-speaking tribes, this family has received relatively little attention in the linguistic literature and many of the individual Siouan languages are severely understudied. This volume aims to make work on Siouan languages more broadly available and to encourage deeper investigation of the myriad typological, theoretical, descriptive, and pedagogical issues they raise. The 17 chapters in this volume present a broad range of current Siouan research, focusing on various Siouan languages, from a variety of linguistic perspectives: historical-genetic, philological, applied, descriptive, formal/generative, and comparative/typological. The editors' preface summarizes characteristic features of the Siouan family, including head-final and "verb-centered" syntax, a complex system of verbal affixes including applicatives and subject-possessives, head-internal relative clauses, gendered speech markers, stop-systems including ejectives, and a preference for certain prosodic and phonotactic patterns. The volume is dedicated to the memory of Professor Robert L. Rankin, a towering figure in Siouan linguistics throughout his long career, who passed away in February of 2014