N. Linial, Y. Mansour, and N. Nisan. Constant depth circuits, Fourier transform, and learnability, J. ACM 40(3):607--620, 1993.  Home/Search   Document Not in Database   Summary   ACM   TOC   Related Articles   Check

.... [6] extend earlier work by Valiant on PAC learning [7, 8] and rely in particular on the concept of VC dimension [9] Work in this area has tended to concentrate on distribution free (i.e. problem independent) analysis although some recent work has given attention to distribution specific analysis [10]. However, even results in distribution specific analysis do not address the issue of the complexity of a specified learning problem. As things stand, then, there is no general method to evaluate the difficulty of a specific learning problem without making assumptions about which learning ....

Linial, N., Mansour, Y. and Nisan, N. (1988). Constant depth circuits, fourier transform and learnability. Proceedings of the 30th I.E.E.E. Symposium on Computational Learning Theory (pp. 56-68). Morgan Kaufmann Publishers.

....the uniform distribution. Thus we regard our work as a (computational) time bounded strengthening of the uncertainty principle. Theoretical Computer Science. There are several relevant areas, and we will distinguish our work from related work in each. First, a relationship was established in [11] between Fourier spectra and learnability. For the class of Boolean functions, the discrete Boolean Fourier basis was defined based on the parity of subsets of the input variables. This representation was used to demonstrate learnability of functions. In [10] authors presented a polynomial time ....

....for this Fourier basis for given Boolean functions. Our work here is related to [10] Clearly their Fourier basis is different from ours (parity of subsets vs trigonometric polynomials) Our basis is the classical one used in convolutions and other well known applications, while the one in [11, 10] proves useful in learning theory context (see [13] for a nice overview of the relationship between their Fourier transforms and complexity theory) Our technique is related to [10] at the high level involving the test of groups of coefficients in order to isolate the large ones, but the technical ....

N. Linial, Y. Mansour, N. Nisan. Constant Depth Circuits, Fourier Transform, and Learnability. JACM 40(3): 607-620 (1993).

....of our paper. We note that our analysis and algorithm is not present in [3] Actually, the purpose of [3] is to study the NNF and its relationship with the other representations. Boolean functions are studied extensively from di erent perspectives coding theory [7, 6] circuit complexity [2, 8] and cryptography [4, 9] are some examples. In all these areas, the Walsh transform is the main tool in the analysis of Boolean functions. However, to the best of our knowledge, the only previously known algorithm for computing the Walsh transform is the fast Walsh transform [1] Hopefully the ....

N. Linial, Y. Mansour and N. Nisan. Constant Depth Circuits, Fourier Transform, and Learnability. Journal of the ACM, 40(3): 607-620 (1993).

.... and construction of Boolean functions for cryptographic applications (see e.g. 11, 16, 19] Moreover several authors have analyzed the Walsh spectrum of Boolean functions and found links between properties of the spectrum and certain computational questions related to the functions (see e.g. [6, 7, 15]) In [1] we showed that the Walsh spectrum of Boolean functions can be analyzed by looking at algebraic properties of a class of Cayley graphs associated with the functions and used this idea to investigate the structure of bernasconi imc.pi.cnr.it codenotti imc.pi.cnr.it Boolean functions ....

....takes the value 1, i.e. f(b(0) f j n , while the other coefficients measure the correlation between the function and the parities of subsets of its arguments. There exists an interesting link between harmonic analysis and circuit complexity of Boolean functions: Linial et al. [15] were able to prove the This transform is also called Abstract Fourier Transform of a Boolean function. following Spectral Lemma, providing a lower bound on the size complexity of Boolean functions computed by constant depth, unbounded fan in, f; g circuits. Lemma 1 ( 15] Let f be a ....

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N. Linial, Y. Mansour, N. Nisan. Constant Depth Circuits, Fourier Transform, and Learnability. Journal of the ACM 40: 607-620 (1993).

....the order of a partition is the number of its de ning features then the magnitude of the Fourier coecients decay exponentially with the order of the corresponding partition; in other words low order coecients are exponentially more signi cant than the higher order coecients. This was proved in [21] for Boolean decision trees. Its counterpart for trees with non Boolean features can be found elsewhere [23] These observations suggest that the spectrum of a decision tree can be approximated by computing only a small number of low order coecients. So Fourier basis o ers an ecient numeric ....

N. Linial, Y. Mansour, and N. Nisan. Constant depth circuits, Fourier transform, and learnability. Journal of the ACM, 40:607-620, 1993.

....algorithm for functions whose spectrum is either sparse or has polynomial L 1 norm. It is easy to show that the L 1 norm of a function is bounded by the number of leaves in any decision tree for that function, even if the nodes may query the parity of arbitrary subsets of the variables. And [LMN89] proves that functions in AC have most of the weight of their spectrum in the coefficients of small sets. These results are used to derive efficient learning algorithms for functions in AC and functions with shallow decision trees. Since OBDD s are such a constrained model of computation, ....

N. Linial, Y. Mansour, and N. Nisan. Constant-depth circuits, Fourier transform, and learnability. Proceedings of the 30th Annual IEEE Symposium on Foundations of Computer Science, October 1989, pp. 574--579.

.... query time is optimal for the Practical RAM, the best current lower bound being the Omega Gamma log n= log log n) one implied by this paper, improving the Omega Gammae 4 log n) one of [Mil96] From circuit complexity there is a result of Mansour, Nisan, and Tiwari [MNT93] building on [LMN93]) that no AC circuit implements a family of universal hash functions fH k g, where implements means that the circuit on input (k; x) computes H k (x) This result is an immediate corollary of our lower bounds as the following argument shows: Suppose, to the contrary, that an AC circuit ....

N. Linial, Y. Mansour, and N. Nisan. Constant depth circuits, Fourier transform, and learnability. Journal of the ACM, 40(3):607--620, July 1993.

....random instances of the subset sum problem can be generated in AC . If subset sum is secure for any length l(n) then our construction gives a one way function and a pseudo random generator computable in AC . This is the first example of such functions. In contrast, Linial, Mansour and Nisan [34] have shown that no pseudo random function generator, as defined by Goldreich, Goldwasser and Micali [20] can be implemented in AC (more precisely, no pseudorandom function secure against an adversary operating in n polylog(n) time can be computable in AC ) We should also note that our ....

N. Linial, Y. Mansour and N. Nisan, Constant depth circuits, Fourier Transform, and Learnability, J. of the ACM., vol. 40, 1993, pp. 607-620.

....recent research. Razborov [13] and Smolensky [14] showed that every AC predicate can be approximated by a low degree polynomial over any finite field. Their upper bounds yield circuit lower bounds related to Hastad s [10] Applying Hastad s lower bound for parity, Linial, Mansour, and Nisan [11] proved an important upper bound: every AC predicate can be approximated by a low degree polynomial over the reals. Their upper bound implies an efficient learning algorithm for AC predicates, under the uniform distribution. An open problem is to efficiently learn AC predicates under an ....

.... for AC can be combined with Smolensky s easy proof that parity cannot be approximated by a low degree polynomial; thus we obtain a very elegant proof that constant depth circuits that compute parity require exponential size (cf. 8, 18, 10, 7, 14] The result of Linial, Mansour, and Nisan [11] could be applied in a similar way, but that would be circular, because their work depends on prior lower bounds for parity. Our work does not depend on prior lower bounds for parity. 2. Perceptrons Studied since the 1950s, perceptrons are depth 2 threshold circuits consisting of a single ....

N. Linial, Y. Mansour, and N. Nisan. Constant depth circuits, Fourier transform, and learnability. In Proceedings of the 30th IEEE Symposium on Foundations of Computer Science, pages 574--579, 1989.

.... It is not hard to demonstrate that the above classes are subclasses of O(log(n) CNF.2 The only nontrivial large depth formulas (depth more than 2) that are proved to be learnable in the literature are read once formulas over different basis [BHH92a, BHH92b, BHHK91] Linial, Mansour and Nisan [LMN93] showed that constant depth circuits are PAC learnable with membership queries in time n poly(log n) under the uniform distribution. The above formula is the first nontrivial class of constant depth formulas that are learnable and have no read restrictions. Result 4. Any boolean function is ....

Nathan Linial, Yishay Mansour and Noam Nisan. Constant depth circuits, Fourier Transform, and Learnability. In Proceedings of the Thirtieth Annual Symposium Foundation of Computer Science, 1989.

....RAM, but not for the AC RAM. The best current lower bound for the Practical RAM is the Omega Gamma log n= log log n) one implied by this paper, improving the Omega Gammae 8 log n) one of [20] From circuit complexity there is a result of Mansour, Nisan, and Tiwari [19] building on [18]) that no AC circuit implements a family of universal hash functions fH k g, where implements means that the circuit on input (k; x) computes H k (x) This result is an immediate corollary of our lower bounds as the following argument shows: Suppose, to the contrary, that circuit computing ....

N. Linial, Y. Mansour, and N. Nisan. Constant depth circuits, Fourier transform, and learnability. Journal of the ACM, 40(3):607--620, July 1993.

....slight sharpening of LMN Johan Hastad Royal Institute of Technology, Stockholm email:johanh nada.kth.se February 18, 2002 Running head: Sharpening of LMN Work done while visiting Institute for Advanced Study, supported by NSF grant CCR 9987077. 1 Abstract Linial, Mansour and Nisan [7] proved that a function computed by a small depth circuit of limited size has most of its Fourier support on small sets. We improve their bounds. When the bottom fanin is bounded we use essentially their argument, but to reduce the general case to this case without a loss in the asymptotic bounds ....

....lemma [5] which says that, if parameters are suitable, such a restriction can enable you to switch a depth two circuit from CNF to DNF without getting a huge blowup. Constant depth circuits are also studied from a learning point of view and a key result here is by Linial, Mansour, and Nisan [7] that proved that such a function can be learned fairly efficiently through the Fourier transform. The key technical result is that a function computed by a small depth circuit of limited size has most of it s Fourier coefficients concentrated on sets of small size. To be more exact, if the ....

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N.Linial, Y.Mansour, N.Nisan. Constant depth circuits, Fourier transform, and learnability. Journal of the ACM 40(3), 607--620 (1993).

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Nathan Linial, Yishay Mansour, and Noam Nisan. Constant depth circuits, Fourier transform, and learnability. Journal of the ACM 40(3):607--620, July 1993.

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N. Linial, Y. Mansour, and N. Nisan. Constant depth circuits, Fourier transform, and learnability, J. ACM 40(3):607--620, 1993.

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N. Linial, Y. Mansour, and N. Nisan. Constant depth circuits, fourier transform, and learnability. Journal of the ACM, 40:607-620, 1993.

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N. Linial, Y. Mansour, N. Nisan. Constant depth circuits, Fourier transform, and learnability. J. Assoc. Comput. Mach. 40, 1993, 607--620.

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Nathan Linial, Yishay Mansour, and Noam Nisan. Constant depth circuits, Fourier transform, and learnability. In 30th Annual Symposium on Foundations of Computer Science, pages 574--579, October 1989.

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N. Linial, Y. Mansour and N. Nisan (1989). Constant depth circuits, Fourier transforms and learnability. In: Proc. of 30th IEEE FOCS, 574--579.

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N. Linial, Y. Mansour, N. Nisan. Constant depth circuits, Fourier transforms, and learnability. Journal of the ACM, 40(3), 1993. 19

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N. Linial, Y. Mansour, N. Nisan. Constant depth circuits, Fourier transform, and learnability. J. of the ACM, vol. 40(3), 1993, pp. 607-620. Prel. version in FOCS'89, pp. 574-579.

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N. Linial, Y. Mansour, and N. Nisan. Constant depth circuits, Fourier transform, and learnability, J. ACM 40(3):607--620, 1993.

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Nathan Linial, Yishay Mansour, and Noam Nisan. Constant depth circuits, fourier transform, and learnability. In Proceedings of the Thirtieth Annual Symposium on Foundations of Computer Science, pages 574-579, Research Triangle Park, North Carolina, October 1989.

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Nathan Linial, Yishay Mansour, Noam Nisan. Constant Depth Circuits, Fourier Transform, and Learnability. In Proceedings of the 31-st Symposium on the Foundations of Computer Science, pp. 574--579, 1989. 394

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Nathan Linial, Yishay Mansour, and Noam Nisan. Constant depth circuits, fourier transform, and learnability. In Proceedings of the Thirtieth Annual Symposium on Foundations of Computer Science, pages 574--579, Research Triangle Park, North Carolina, October 1989.

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Linial, N., Mansour, Y., and Nisan, N. (1993). Constant depth circuits, Fourier transforms, and learnability. Journal of the ACM 40, 607--620.

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