Related papers: On (not) learning the M\"obius function
We study pseudo-Abelian integrals associated with polynomial deformations of slow-fast Darboux integrable systems. Under some assumptions we prove local boundedness of the number of their zeros.
Any function F : {0,...,N-1} -> {-1,1} such that F(x) can be computed from the binary digits of x using a bounded depth circuit is orthogonal to the Mobius function mu in the sense that E_{0 <= x <= N-1} mu(x)F(x) = o(1). The proof combines…
We propose a family of quantum algorithms for estimating Gowers uniformity norms $ U^k $ over finite abelian groups and demonstrate their applications to testing polynomial structure and counting arithmetic progressions. Building on recent…
Building on work of Charton, we train small transformer models to calculate the M\"{o}bius function $\mu(n)$ and the squarefree indicator function $\mu^2(n)$. The models attain nontrivial predictive power. We apply a mixture of additional…
In this paper we describe an algorithm for the computation of canonical forms of finite subsets of $\mathbb{Z}^d$, up to affinities over $\mathbb{Z}$. For fixed dimension $d$, this algorithm has worst-case asymptotic complexity $O(n \log^2…
Recent advances have demonstrated that $\mathcal{O}(\log M)$ measurements suffice to predict $M$ properties of arbitrarily large quantum many-body systems. However, these remarkable findings assume that the properties to be predicted are…
A large set of signals can sometimes be described sparsely using a dictionary, that is, every element can be represented as a linear combination of few elements from the dictionary. Algorithms for various signal processing applications,…
We consider learning under the constraint of local differential privacy (LDP). For many learning problems known efficient algorithms in this model require many rounds of communication between the server and the clients holding the data…
In this paper we study the approximate learnability of valuations commonly used throughout economics and game theory for the quantitative encoding of agent preferences. We provide upper and lower bounds regarding the learnability of…
We review and numerically study a protocol for Liouvillian learning based on randomized Pauli states and measurements. In particular, in the two-body, long-range interactions, and single-body noise setting, we describe the complete workflow…
The correlation functions for models of minimal gravity are discussed. An algorithm is proposed for calculations of invariant ratios from formulas of residues that can be compared with the coefficients of expansion of the partition function…
Research on the distribution of prime numbers has revealed a dual character: deterministic in definition yet exhibiting statistical behavior reminiscent of random processes. In this paper we show that it is possible to use an image-focused…
We prove a Liouville type theorem for entire maximal $m$-subharmonic functions in $\mathbb C^n$ with bounded gradient. This result, coupled with a standard blow-up argument, yields a (non-explicit) a priori gradient estimate for the complex…
We present new differentially private algorithms for learning a large-margin halfspace. In contrast to previous algorithms, which are based on either differentially private simulations of the statistical query model or on private convex…
The polynomial method and the Ambainis's lower bound (or \emph{Alb}, for short) method are two main quantum lower bound techniques. While recently Ambainis showed that the polynomial method is not tight, the present paper aims at studying…
Using a sieve procedure akin to the sieve of Eratosthenes we show how for each prime $p$ to build the corresponding M\"obius prime-function, which in the limit of infinitely large primes becomes identical to the original M\"obius function.…
We show that for any $\varepsilon > 0$, prime $q$ sufficiently large with respect to $1 / \varepsilon$ and residue class $(a,q) = 1$, there exist two integers $m, n \leq q^{5/2 + \varepsilon}$ with $m \equiv n \equiv a \pmod{q}$ such that…
Minimum Description Length (MDL) is an important principle for induction and prediction, with strong relations to optimal Bayesian learning. This paper deals with learning non-i.i.d. processes by means of two-part MDL, where the underlying…
We study a number of filtering schemes for the reduction of the statistical error in non-adiabatic calculations by means of the quantum-classical Liouville equation. In particular, we focus on a scheme based on setting a threshold value on…
In the Bayesian approach to sequential decision making, exact calculation of the (subjective) utility is intractable. This extends to most special cases of interest, such as reinforcement learning problems. While utility bounds are known to…