Notes on the Shannon Entropy of the Neural Response (2009)
Shakhnarovich, Greg, Bouvrie, Jake, Rosasco, Lorenzo, Smale, Steve
In these notes we focus on the concept of Shannon entropy in an attempt to provide a systematic way of assessing the discrimination properties of the neural response, and quantifying the role played...
P, NP and mathematics – a computational complexity (2008)
Avi Wigderson, Steve Smale, Avi Wigderson
perspective
Mathematics of the Neural Response (2008)
Smale, Steve, Rosasco, Lorenzo, Bouvrie, Jake, Caponnetto, Andrea, Poggio, Tomaso
We propose a natural image representation, the neural response, motivated by the neuroscience of the visual cortex. The inner product defined by the neural response leads to a similarity measure...
Mathematics of the Neural Response (2008)
Caponnetto, Andrea, Poggio, Tomaso, Bouvrie, Jake, Rosasco, Lorenzo, Smale, Steve
We propose a natural image representation, the neural response, motivated by the neuroscience of the visual cortex. The inner product defined by the neural response leads to a similarity measure...
On a model of visual cortex: learning invariance and selectivity (2008)
Caponnetto, Andrea, Poggio, Tomaso, Smale, Steve
In this paper we present a class of algorithms for similarity learning on spaces of images. The general framework that we introduce is motivated by some well-known hierarchical pre-processing...
On a model of visual cortex: learning invariance and selectivity (2008)
Caponnetto, Andrea, Poggio, Tomaso, Smale, Steve
In this paper we present a class of algorithms for similarity learning on spaces of images. The general framework that we introduce is motivated by some well-known hierarchical pre-processing...
Peter Bartlett, Evarist Giné, Vladimir Koltchinskii, Gábor Lugosi, Shahar Mendelson, Vitali Milman, ...
Berkeley) Acknowledgements. The Mathematical Foundations of Learning Theory is supported by the “Ministerio de Ciencia y Tecnología ” (ref. BFM2002-11039-E).
NP and Mathematics - a computational complexity perspective (2008)
“P versus N P – a gift to mathematics from Computer Science”
Mathematics of the neural response (2008)
Steve Smale, Lorenzo Rosasco, Jake Bouvrie, Andrea Caponnetto
We propose a natural image representation, the neural response, motivated by the neuroscience of the visual cortex. The inner product defined by the neural response leads to a similarity measure...
Algebraic Settings for the Problem "P != NP?" (2007)
Lenore Blum, Felipe Cucker, Mike Shub, Steve Smale
. When complexity theory is studied over an arbitrary unordered field K, the classical theory is recaptured with K = Z 2 . The fundamental result that the Hilbert Nullstellensatz as a decision...
Online Learning with Markov Sampling ∗ (2007)
This paper attempts to give an extension of learning theory to a setting where the assumption
Derived Distance: towards a mathematical theory of visual cortex. (2007)
Steve Smale, Tomaso Poggio, Andrea Caponnetto, Jake Bouvrie
We describe a “natural ” metric on the space of images motivated by the neuroscience of visual cortex. We propose the notion of a hierarchical derived distance and suggest that it could be...
On the mathematics of emergence (2006)
A common situation occurring in a number of disciplines is that in which a number of autonomous agents reach a consensus without a central direction. An example of this is the emergence of a common...
Online learning algorithms (2005)
In this paper, we study an online learning algorithm in Reproducing Kernel Hilbert Spaces (RKHS) and general Hilbert spaces. We present a general form of the stochastic gradient method to minimize a...
The mathematics of learning: Dealing with data (2003)
Draft for the Notices of the AMS Learning is key to developing systems tailored to a broad range of data analysis and information extraction tasks. We outline the mathematical foundations of learning...
The Mathematics of Learning: Dealing with Data (2003)
Learning is key to developing systems tailored to a broad range of data analysis and information extraction tasks. We outline the mathematical foundations of learning theory and describe a key...
On the mathematical foundations of learning (2002)
The problem of learning is arguably at the very core of the problem of intelligence, both biological and arti cial. T. Poggio and C.R. Shelton
On the mathematical foundations of learning (2002)
The problem of learning is arguably at the very core of the problem of intelligence, both biological and artificial. T. Poggio and C.R. Shelton
0 Preface Complexity Theory and Numerical Analysis ∗ (2000)
Complexity theory of numerical analysis is the study of the number of arithmetic operations required to pass from the input to the output of a numerical problem. To a large extent this requires the...
A Polynomial Time Algorithm for Diophantine Equations in One Variable (1999)
Felipe Cucker, Pascal Koiran, Steve Smale
The goal of this paper is to prove the following. Theorem 1 There is a polynomial time algorithm which given input f 2 ZZ[t] decides
A Polynomial Time Algorithm for Diophantine Equations in One Variable (1999)
Felipe Cucker, Pascal Koiran, Steve Smale
this paper is to prove the following.
Mathematical Problems for the Next Century 1 Second Version (1998)
Introduction. V. I. Arnold, on behalf of the International Mathematical Union has written to a number of mathematicians with a suggestion that they describe some great problems for the next century....
A polynomial time algorithm for diophantine equations in one variable. (1997)
Cucker, Felipe, Koiran, Pascal, Smale, Steve
(eng) We show that the integer roots of of a univariate polynomial with integer coefficients can be computed in polynomial time. This result holds for the classical (i.e. Turing) model of computation...
APolynomial Time Algorithm for Diophantine Equations in One Variable (1997)
Felipe Cucker, Pascal Koiran, Steve Smale, Felipe Cucker, Pascal Koiran, Steve Smale, ...
We show that the integer roots of of a univariate polynomial with integer coe cients can be computed in polynomial time. This result holds for the classical (i.e. Turing) model of computation and a...
Complexity theory and numerical analysis (1997)
Complexity theory of numerical analysis is the study of the number of arithmetic operations required to pass from the input to the output of a numerical problem. To a large extent this requires the...
Complexity theory and numerical analysis (1997)
Complexity theory of numerical analysis is the study of the number of arithmetic operations required to pass from the input to the output of a numerical problem. To a large extent this requires the...
On the intractability of Hilbert's Nullstellensatz and an algebraic version (1996)
In this paper we relate an elementary problem in number theory to the intractability of deciding whether an algebraic set defined over the complex numbers (or any algebraically closed field of...
On the intractability of Hilbert's Nullstellensatz and an algebraic version (1996)
In this paper we relate an elementary problem in number theory to the intractability of deciding whether an algebraic set defined over the complex numbers (or any algebraically closed field of...
Complexity of Bezout’s theorem. IV. Probability of success; extensions (1996)
Abstract. We estimate the probability that a given number of projective Newton steps applied to a linear homotopy of a system of n homogeneous polynomial equations in n + 1 complex variables of fixed...
Complexity and Real Computation: A Manifesto (1995)
Lenore Blum, Felipe Cucker, Mike Shub, Steve Smale
. Finding a natural meeting ground between the highly developed complexity theory of computer science ---with its historical roots in logic and the discrete mathematics of the integers--- and the...
Smale: Complexity of Bezout's theorem V: Polynomial time (1994)
The main goal of this paper is to show that the problem of finding approximately a zero of a polynomial system of equations can be solved in polynomial time,