From bounded moments to certifiable moments with limited orders

Establish whether distributions on R^d with (m, B_m)-bounded moments, where m = polylog(d) and B_m = poly(m), admit sum-of-squares certificates for bounded fourth moments with certifiable bound B'_4 = poly(m); specifically, show whether there exist degree-m'' (possibly larger than m) sum-of-squares proofs (with or without the unit-norm axiom ||v||_2^2 = 1) certifying that C·B'_4^4 ||v||_2^4 − E_{X}[⟨v, X⟩^4] ≥ 0 for all v, with B'_4 polynomial in m.

Background

The paper highlights a broader program: relating moment-boundedness to certifiable boundedness via sum-of-squares. Hardness reductions under the Small Set Expansion Hypothesis show that, for constant m, certifiability with dimension-independent bounds is impossible in general. However, the regime where m grows slowly with dimension—e.g., m = polylog(d)—may be tractable, and would meaningfully expand the class of distributions admitting useful SoS certificates.

Clarifying whether polylogarithmic moment bounds suffice to certify fourth moments with polynomial bounds would bridge a key gap between information-theoretic moment assumptions and algorithmically verifiable certificates, with implications for robust estimation tasks where fourth-moment certifiability is pivotal.