no code implementations • 5 Mar 2022 • Samuel B. Hopkins, Tselil Schramm, Jonathan Shi
We give a spectral algorithm for decomposing overcomplete order-4 tensors, so long as their components satisfy an algebraic non-degeneracy condition that holds for nearly all (all but an algebraic set of measure $0$) tensors over $(\mathbb{R}^d)^{\otimes 4}$ with rank $n \le d^2$.
no code implementations • 6 Oct 2016 • Tengyu Ma, Jonathan Shi, David Steurer
We give new algorithms based on the sum-of-squares method for tensor decomposition.
no code implementations • 8 Dec 2015 • Samuel B. Hopkins, Tselil Schramm, Jonathan Shi, David Steurer
For tensor decomposition, we give an algorithm with running time close to linear in the input size (with exponent $\approx 1. 086$) that approximately recovers a component of a random 3-tensor over $\mathbb R^n$ of rank up to $\tilde \Omega(n^{4/3})$.
no code implementations • 12 Jul 2015 • Samuel B. Hopkins, Jonathan Shi, David Steurer
We study a statistical model for the tensor principal component analysis problem introduced by Montanari and Richard: Given a order-$3$ tensor $T$ of the form $T = \tau \cdot v_0^{\otimes 3} + A$, where $\tau \geq 0$ is a signal-to-noise ratio, $v_0$ is a unit vector, and $A$ is a random noise tensor, the goal is to recover the planted vector $v_0$.