We present a deterministic $2^{k^{\mathcal{O}(1)}} \text{poly}(n,d)$ time algorithm for decomposing $d$-dimensional, width-$n$ tensors of rank at most $k$ over $\mathbb{R}$ and $\mathbb{C}$. This improves upon the previous randomized algorithm of Peleg, Shpilka, and Volk (ITCS '24) that takes $2^{k^{k^{\mathcal{O}(k)}}} \text{poly}(n,d)$ time and the deterministic $n^{k^k}$ time algorithms of Bhargava, Saraf, ... more >>>

We present a new algorithm for solving homogeneous multilinear equations, which are high dimensional generalisations of solving homogeneous linear equations. First, we present a linear time reduction from solving generic homogeneous multilinear equations to computing hyperdeterminants, via a high dimensional Cramer's rule. Hyperdeterminants are generalisations of determinants, associated with tensors ... more >>>

Consider the model where we can access a parity function through random uniform labeled examples in the presence of random classification noise. In this paper, we show that approximating the number of relevant variables in the parity function is as hard as properly learning parities.

More specifically, let $\gamma:{\mathbb R}^+\to ... more >>>