Journal ArticleDOI
Constant depth circuits, Fourier transform, and learnability
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TLDR
It is shown that an ACO Boolean function has almost all of its "power spectrum" on the low-order coefficients, implying several new properties of functions in -4C(': Functions in AC() have low "average sensitivity;" they may be approximated well by a real polynomial of low degree and they cannot be pseudorandom function generators.Abstract:
In this paper, Boolean functions in ,4C0 are studied using harmonic analysis on the cube. The main result is that an ACO Boolean function has almost all of its "power spectrum" on the low-order coefficients. An important ingredient of the proof is Hastad's switching lemma (8). This result implies several new properties of functions in -4C(': Functions in AC() have low "average sensitivity;" they may be approximated well by a real polynomial of low degree and they cannot be pseudorandom function generators. Perhaps the most interesting application is an O(n POIYIOg(n ')-time algorithm for learning func- tions in ACO. The algorithm observes the behavior of an AC'" function on O(nPO'Y'Og(n)) randomly chosen inputs, and derives a good approximation for the Fourier transform of the function. This approximation allows the algorithm to predict, with high probability, the value of the function on other randomly chosen inputs.read more
Citations
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Scaling learning algorithms towards AI
TL;DR: It is argued that deep architectures have the potential to generalize in non-local ways, i.e., beyond immediate neighbors, and that this is crucial in order to make progress on the kind of complex tasks required for artificial intelligence.
Book ChapterDOI
Boolean Models and Methods in Mathematics, Computer Science, and Engineering: Boolean Functions for Cryptography and Error-Correcting Codes
TL;DR: Encryption-decryption is the most ancient cryptographic activity, but its nature has deeply changed with the invention of computers, because the cryptanalysis (the activity of the third person, the eavesdropper, who aims at recovering the message) can use their power.
Book
Analysis of Boolean Functions
TL;DR: This text gives a thorough overview of Boolean functions, beginning with the most basic definitions and proceeding to advanced topics such as hypercontractivity and isoperimetry, and includes a "highlight application" such as Arrow's theorem from economics.
Proceedings ArticleDOI
Delegating computation: interactive proofs for muggles
TL;DR: This work shows how to construct short (polylog size) computationally sound non-interactive certificates of correctness for any log-space uniform NC computation, in the public-key model, and settles an open question regarding the expressive power of proof systems with such verifiers.
References
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Proceedings ArticleDOI
A theory of the learnable
TL;DR: This paper regards learning as the phenomenon of knowledge acquisition in the absence of explicit programming, and gives a precise methodology for studying this phenomenon from a computational viewpoint.
Journal ArticleDOI
How to construct random functions
TL;DR: In this paper, a constructive theory of randomness for functions, based on computational complexity, is developed, and a pseudorandom function generator is presented, which is a deterministic polynomial-time algorithm that transforms pairs (g, r), where g is any one-way function and r is a random k-bit string, to computable functions.
Book ChapterDOI
How to construct random functions
TL;DR: A constructive theory of randomness for functions, based on computational complexity, is developed, and a pseudorandom function generator is presented that has applications in cryptography, random constructions, and complexity theory.
Proceedings ArticleDOI
Theory and application of trapdoor functions
TL;DR: A new information theory is introduced and the concept of trapdoor functions is studied and applications of such functions in cryptography, pseudorandom number generation, and abstract complexity theory are examined.
Proceedings ArticleDOI
Algebraic methods in the theory of lower bounds for Boolean circuit complexity
TL;DR: It is proved that depth k circuits with gates NOT, OR and MODp where p is a prime require Exp(&Ogr;(n1/2k)) gates to calculate MODr functions for any r ≠ pm.