We show that there exists a class of boolean functions C such that $(i)$ there is a distribution-independent statistical query algorithm for learning C that makes a polynomial number of queries of inverse polynomial tolerance and $(ii)$ for any set of functions $\Phi_1, \dots, \Phi_r$ such that for all $f \in$ C we can write $f(x) = \text{sign} \left( \sum_{i = 1}^r w_i \Phi_i(x) \right)$ for some set of weights $w_i \in \mathbb{R}$, we must have that $r \geq n^{\omega(1)}$. This gives a superpolynomial separation between dimension complexity and the query complexity of distribution-free learning in the statistical query model, negatively answering a question of Feldman, Kamath, and Srebro [FKS26]. Our construction C is a subclass of DNFs, and the proof is a simple consequence of recent progress on agnostically learning conjunctions [DKR25,CPS26] and the work of Razborov and Sherstov on the sign rank of DNFs [RS10].
A degree-$d$ polynomial source is the output of a polynomial map of degree at most $d$ over $\mathbb{F}_2$ on arbitrarily many uniform random bits. Khodabandeh and Shinkar (FOCS'26) proved that $\mathrm{Ber}(1/3)^{\otimes N}$ has statistical distance $1-o(1)$ from every constant-degree polynomial source and conjectured...
Consider a constraint satisfaction problem $\mathbf{C}$ on finitely many independent random variables with dependency graph $G$. Let $p_a$ be the violation probability of a constraint $a\in \mathbf{C}$ and $N_G^2 (a)$ the set of constraints at distance one or two from $a$ in $G$. Suppose that, there exists $x\in (0,1)^...
Vishesh Jain, Clayton Mizgerd, H. Pham· 0 citations
If each coordinate appears in only a small number of clauses, there is a quantum algorithm for strongly log-concave sampling using local queries using $\widetilde{O}(\sqrt{\kappa}d)$ local queries, where $\kappa$ is the condition number.
Let $X_1,\ldots,X_n$ be almost surely distinct independent real-valued random variables. Let $\sigma$ be the random permutation of $\{1,\ldots,n\}$ such that $X_{\sigma(1)}<X_{\sigma(2)}<\cdots<X_{\sigma(n)}$. We show that the set of laws of $\sigma$, as the laws of $X_1,\ldots,X_n$ vary, has semialgebraic dimension \[...
We give an approximately uniform sampler for satisfying assignments of constraint satisfaction problems that satisfy $4\mathrm e p(\Delta+1)^2\le1$, where $p$ is the largest constraint-violation probability under the uniform product distribution, and $\Delta$ is the maximum degree of the dependency graph. The algorithm...
A boolean function $f$ has Fourier dimension $k$ if its nonzero Fourier coefficients span a subspace of dimension $k$. We consider the property testing task of determining whether a function has Fourier dimension at most $k$, or is $\epsilon$-far from being so. We show that there is a $O(k/\sqrt{\epsilon})$-query quant...
Kenny Chen· 0 citations
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