### Simulation and Retargeting of Complex Multi-Character Interactions

Yunbo Zhang, Deepak Gopinath, Yuting Ye, Jessica Hodgins, Greg Turk, Jungdam Won

International Conference on Learning Representations (ICLR)

We consider the problem of learning a one-hidden-layer neural network: we assume the input x ∈ R^{d} is from Gaussian distribution and the label y = a^{T}σ(Bx) + ξ, where a is a nonnegative vector in R m with m ≤ d, B ∈ R^{m×d} is a full-rank weight matrix, and ξ is a noise vector. We first give an analytic formula for the population risk of the standard squared loss and demonstrate that it implicitly attempts to decompose a sequence of low-rank tensors simultaneously. Inspired by the formula, we design a non-convex objective function *G*(·) whose landscape is guaranteed to have the following properties:

- All local minima of
*G*are also global minima. - All global minima of
*G*correspond to the ground truth parameters. - The value and gradient of
*G*can be estimated using samples.

With these properties, stochastic gradient descent on *G* provably converges to the global minimum and learn the ground-truth parameters. We also prove finite sample complexity results and validate the results by simulations.

Yunbo Zhang, Deepak Gopinath, Yuting Ye, Jessica Hodgins, Greg Turk, Jungdam Won

Harrison Jesse Smith, Qingyuan Zheng, Yifei Li, Somya Jain, Jessica K. Hodgins

Simran Arora, Patrick Lewis, Angela Fan, Jacob Kahn, Christopher Ré