An Orthogonal Basis for Functions over a Slice of the Boolean Hypercube

Filmus, Yuval

doi:10.37236/4567

Cited by 38 publications

(74 citation statements)

References 16 publications

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“…Proof. The first two statements are the content of Lemma 4.3 in [Fil16]. The dimension of Y m is given in Lemma 2.1 in [Fil16].…”

Section: A1 Structure Of Xmentioning

confidence: 98%

“…While this may seem to render recovery hopeless at first glance, this is not the case, due to the fact that many eigenvectors (actually, eigenspaces) of X contain non-trivial information about the spike x * , as opposed to only the top one. We prove this by exploiting the special structure of X through the Johnson scheme, and using tools from Fourier analysis on a slice of the hypercube, in particular a Poincaré-type inequality by [Fil16].…”

Section: Compute a (Unit-norm) Leading Eigenvectormentioning

confidence: 99%

“…Our contribution can be viewed as showing that even when we restrict to the submatrix M of M supported on subsets of size ℓ, the leading eigenvector still allows us to recover x * whenever the SNR is sufficiently large. Proving this requires us to perform Fourier analysis over a slice of the hypercube rather than the simpler setting of the entire hypercube, which we do by appealing to Johnson schemes and some results of [Fil16].…”

Section: Intuition: Higher-order Message-passing and Maximum Likelihoodmentioning

confidence: 99%

“…Filmus [Fil16] provides a common eigenbasis for this algebra: for 0 ≤ m ≤ ℓ, let ϕ = (a 1 , b 1 , . .…”

Section: A1 Structure Of Xmentioning

confidence: 99%

“…To prove Lemma A.10, we need some results on Fourier analysis on the slice of the hypercube [n] ℓ . Following [Fil16], we define the following. First, given a function f : [n] ℓ → R, we define its expectation as its average value over all sets of size ℓ, and write We also define its variance as…”

Section: A34 a Poincaré Inequality On A Slice Of The Hypercubementioning

confidence: 99%

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The Kikuchi Hierarchy and Tensor PCA

Wein

Alaoui

Moore

2019

2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)

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For the tensor PCA (principal component analysis) problem, we propose a new hierarchy of increasingly powerful algorithms with increasing runtime. Our hierarchy is analogous to the sumof-squares (SOS) hierarchy but is instead inspired by statistical physics and related algorithms such as belief propagation and AMP (approximate message passing). Our level-ℓ algorithm can be thought of as a linearized message-passing algorithm that keeps track of ℓ-wise dependencies among the hidden variables. Specifically, our algorithms are spectral methods based on the Kikuchi Hessian, which generalizes the well-studied Bethe Hessian to the higher-order Kikuchi free energies.It is known that AMP, the flagship algorithm of statistical physics, has substantially worse performance than SOS for tensor PCA. In this work we 'redeem' the statistical physics approach by showing that our hierarchy gives a polynomial-time algorithm matching the performance of SOS. Our hierarchy also yields a continuum of subexponential-time algorithms, and we prove that these achieve the same (conjecturally optimal) tradeoff between runtime and statistical power as SOS. Our proofs are much simpler than prior work, and also apply to the related problem of refuting random k-XOR formulas. The results we present here apply to tensor PCA for tensors of all orders, and to k-XOR when k is even.Our methods suggest a new avenue for systematically obtaining optimal algorithms for Bayesian inference problems, and our results constitute a step toward unifying the statistical physics and sum-of-squares approaches to algorithm design. *

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“…Proof. The first two statements are the content of Lemma 4.3 in [Fil16]. The dimension of Y m is given in Lemma 2.1 in [Fil16].…”

Section: A1 Structure Of Xmentioning

confidence: 98%

Section: Compute a (Unit-norm) Leading Eigenvectormentioning

confidence: 99%

Section: Intuition: Higher-order Message-passing and Maximum Likelihoodmentioning

confidence: 99%

“…Filmus [Fil16] provides a common eigenbasis for this algebra: for 0 ≤ m ≤ ℓ, let ϕ = (a 1 , b 1 , . .…”

Section: A1 Structure Of Xmentioning

confidence: 99%

Section: A34 a Poincaré Inequality On A Slice Of The Hypercubementioning