Resilience of Rademacher chaos of low degree

The {\em resilience} of a Rademacher chaos is the maximum number of adversarial sign-flips that the chaos can sustain without having its largest atom probability significantly altered. Inspired by probabilistic lower-bound guarantees for the resilience of linear Rademacher chaos (aka. resilience of the Littlewood-Offord problem), obtained by Bandeira, Ferber, and Kwan (Advances in Mathematics, Vol. $319$, $2017$), we provide probabilistic lower-bound guarantees for the resilience of Rademacher chaos of arbitrary degree; these being most meaningful provided that the degree is constant.

💡 Research Summary

The paper introduces and studies the notion of resilience for Rademacher chaos, i.e., multilinear polynomials evaluated on independent Rademacher vectors. Resilience is defined as the minimal Hamming distance that must be altered in the input signs to force the polynomial to take a prescribed value x; equivalently, it measures how many adversarial sign‑flips the chaos can tolerate before its largest atom probability changes significantly.

The authors begin by recalling the classical Littlewood‑Offord problem and recent work on the resilience of linear Rademacher chaos (Bandeira, Ferber, Kwan, 2017), which showed that for any non‑zero coefficient vector a, the resilience is Ω(log log n) with high probability. They then ask whether similar guarantees hold for higher‑degree chaos, especially when the degree d is a fixed constant.

Two main results are presented. The first is a model theorem (Theorem 1.2) for bilinear (degree‑2) chaos of the form ψᵀMξ, where M∈ℝ^{n×m} and ψ, ξ are independent Rademacher vectors. The theorem introduces several matrix parameters: the entrywise infinity norm ‖M‖∞, the maximal row ℓ₂‑norm ‖M‖{∞,2}, the maximal row support size ‖M‖_{∞,0}, the Frobenius norm ‖M‖_F, and the stable rank sr(M)=‖M‖_F²/‖M‖_2². Using these, two auxiliary quantities f(M,n) and g(M,r) are defined, capturing sparsity and diameter effects. The result states that for any integer r, \