528 research outputs found
Sign rank versus VC dimension
This work studies the maximum possible sign rank of sign
matrices with a given VC dimension . For , this maximum is {three}. For
, this maximum is . For , similar but
slightly less accurate statements hold. {The lower bounds improve over previous
ones by Ben-David et al., and the upper bounds are novel.}
The lower bounds are obtained by probabilistic constructions, using a theorem
of Warren in real algebraic topology. The upper bounds are obtained using a
result of Welzl about spanning trees with low stabbing number, and using the
moment curve.
The upper bound technique is also used to: (i) provide estimates on the
number of classes of a given VC dimension, and the number of maximum classes of
a given VC dimension -- answering a question of Frankl from '89, and (ii)
design an efficient algorithm that provides an multiplicative
approximation for the sign rank.
We also observe a general connection between sign rank and spectral gaps
which is based on Forster's argument. Consider the adjacency
matrix of a regular graph with a second eigenvalue of absolute value
and . We show that the sign rank of the signed
version of this matrix is at least . We use this connection to
prove the existence of a maximum class with VC
dimension and sign rank . This answers a question
of Ben-David et al.~regarding the sign rank of large VC classes. We also
describe limitations of this approach, in the spirit of the Alon-Boppana
theorem.
We further describe connections to communication complexity, geometry,
learning theory, and combinatorics.Comment: 33 pages. This is a revised version of the paper "Sign rank versus VC
dimension". Additional results in this version: (i) Estimates on the number
of maximum VC classes (answering a question of Frankl from '89). (ii)
Estimates on the sign rank of large VC classes (answering a question of
Ben-David et al. from '03). (iii) A discussion on the computational
complexity of computing the sign-ran
Consistency-Checking Problems: A Gateway to Parameterized Sample Complexity
Recently, Brand, Ganian and Simonov introduced a parameterized refinement of
the classical PAC-learning sample complexity framework. A crucial outcome of
their investigation is that for a very wide range of learning problems, there
is a direct and provable correspondence between fixed-parameter
PAC-learnability (in the sample complexity setting) and the fixed-parameter
tractability of a corresponding "consistency checking" search problem (in the
setting of computational complexity). The latter can be seen as generalizations
of classical search problems where instead of receiving a single instance, one
receives multiple yes- and no-examples and is tasked with finding a solution
which is consistent with the provided examples.
Apart from a few initial results, consistency checking problems are almost
entirely unexplored from a parameterized complexity perspective. In this
article, we provide an overview of these problems and their connection to
parameterized sample complexity, with the primary aim of facilitating further
research in this direction. Afterwards, we establish the fixed-parameter
(in)-tractability for some of the arguably most natural consistency checking
problems on graphs, and show that their complexity-theoretic behavior is
surprisingly very different from that of classical decision problems. Our new
results cover consistency checking variants of problems as diverse as (k-)Path,
Matching, 2-Coloring, Independent Set and Dominating Set, among others
Computational and Robotic Models of Early Language Development: A Review
We review computational and robotics models of early language learning and
development. We first explain why and how these models are used to understand
better how children learn language. We argue that they provide concrete
theories of language learning as a complex dynamic system, complementing
traditional methods in psychology and linguistics. We review different modeling
formalisms, grounded in techniques from machine learning and artificial
intelligence such as Bayesian and neural network approaches. We then discuss
their role in understanding several key mechanisms of language development:
cross-situational statistical learning, embodiment, situated social
interaction, intrinsically motivated learning, and cultural evolution. We
conclude by discussing future challenges for research, including modeling of
large-scale empirical data about language acquisition in real-world
environments.
Keywords: Early language learning, Computational and robotic models, machine
learning, development, embodiment, social interaction, intrinsic motivation,
self-organization, dynamical systems, complexity.Comment: to appear in International Handbook on Language Development, ed. J.
Horst and J. von Koss Torkildsen, Routledg
Protecting Temporal Fingerprints with Synchronized Chaotic Circuits
In recent years, connected autonomous vehicles (CAVs) feature an increasing number of Ethernet-enabled electronic control units (ECUs), thereby creating more threat vectors that provide access to the Controller Area Network (CAN) Bus. Currently, mitigation techniques to protect the CAN bus from compromised ECU units in vehicle ad hoc networks (VANET) often utilize classical cryptographic techniques. However, ECUs often have temporal signatures that leak internal state information to eavesdropping attackers who can leverage temporal properties for longitudinal attacks. Unfortunately, these types of attacks are difficult to defend against using classical encryption schemes and intrusion detection systems (IDS) due to their high computational demands and ineffectiveness at protecting CAVs throughout the duration of their long lifespans. In order to address these problems, we propose a novel cryptographic framework that protects information embedded in ECU network communications by delivering an encryption system that periodically salts the temporal dynamics of individual ECU units with chaotic signals that are difficult to learn. We demonstrate the framework on two datasets, and our results show that the underlying temporal signatures cannot be approximated by state-of-the-art learning algorithms over finite time horizons
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