On Extractors and Exposure-Resilient Functions for Sublogarithmic Entropy

We study deterministic extractors for oblivious bit-fixing sources (a.k.a. resilient functions) and exposure-resilient functions with small min-entropy: of the function's n input bits, k << n bits are uniformly random and unknown to the adversary. We simplify and improve an explicit construction of extractors for bit-fixing sources with sublogarithmic k due to Kamp and Zuckerman (SICOMP 2006), achieving error exponentially small in k rather than polynomially small in k. Our main result is that when k is sublogarithmic in n, the short output length of this construction (O(log k) output bits) is optimal for extractors computable by a large class of space-bounded streaming algorithms. Next, we show that a random function is an extractor for oblivious bit-fixing sources with high probability if and only if k is superlogarithmic in n, suggesting that our main result may apply more generally. In contrast, we show that a random function is a static (resp. adaptive) exposure-resilient function with high probability even if k is as small as a constant (resp. log log n). No explicit exposure-resilient functions achieving these parameters are known

On Resilient and Exposure-Resilient Functions

Resilient and exposure-resilient functions are functions whose output appears random even if some portion of their input is either revealed or fixed. We explore an alternative way of characterizing these objects that ties them explicitly to the theory of randomness extractors and simplifies current proofs of basic results. We also describe the inclusions and separations governing the various classes of resilient and exposure-resilient functions. Using this knowledge, we explore the possibility of improving existing constructions of these functions and prove that one specific method of doing so is impossible

On Extractors and Exposure-Resilient Functions for Sublogarithmic Entropy

We study resilient functions and exposure-resilient functions in the low-entropy regime. A resilient function (a.k.a. deterministic extractor for oblivious bit-fixing sources) maps any distribution on n -bit strings in which k bits are uniformly random and the rest are fixed into an output distribution that is close to uniform. With exposure-resilient functions, all the input bits are random, but we ask that the output be close to uniform conditioned on any subset of n - k input bits. In this paper, we focus on the case that k is sublogarithmic in n. We simplify and improve an explicit construction of resilient functions for k sublogarithmic in n due to Kamp and Zuckerman (SICOMP 2006), achieving error exponentially small in k rather than polynomially small in k. Our main result is that when k is sublogarithmic in n, the short output length of this construction (O(log k) output bits) is optimal for extractors computable by a large class of space-bounded streaming algorithms. Next, we show that a random function is a resilient function with high probability if and only if k is superlogarithmic in n, suggesting that our main result may apply more generally. In contrast, we show that a random function is a static (resp. adaptive) exposure-resilient function with high probability even if k is as small as a constant (resp. loglog n). No explicit exposure-resilient functions achieving these parameters are known.Engineering and Applied SciencesMathematic

Functionally informed fine-mapping and polygenic localization of complex trait heritability

Fine-mapping aims to identify causal variants impacting complex traits. We propose PolyFun, a computationally scalable framework to improve fine-mapping accuracy by leveraging functional annotations across the entire genome-not just genome-wide-significant loci-to specify prior probabilities for fine-mapping methods such as SuSiE or FINEMAP. In simulations, PolyFun + SuSiE and PolyFun + FINEMAP were well calibrated and identified >20% more variants with a posterior causal probability >0.95 than identified in their nonfunctionally informed counterparts. In analyses of 49 UK Biobank traits (average n = 318,000), PolyFun + SuSiE identified 3,025 fine-mapped variant-trait pairs with posterior causal probability >0.95, a >32% improvement versus SuSiE. We used posterior mean per-SNP heritabilities from PolyFun + SuSiE to perform polygenic localization, constructing minimal sets of common SNPs causally explaining 50% of common SNP heritability; these sets ranged in size from 28 (hair color) to 3,400 (height) to 2 million (number of children). In conclusion, PolyFun prioritizes variants for functional follow-up and provides insights into complex trait architectures. PolyFun is a computationally scalable framework for functionally informed fine-mapping that makes full use of genome-wide data. It prioritizes more variants than previous methods when applied to 49 complex traits from UK Biobank.Peer reviewe