On the growth of von Neumann dimension of harmonic spaces of semipositive line bundles over covering manifolds

We study the harmonic space of line bundle valued forms over a covering manifold with a discrete group action $\Gamma$, and obtain an asymptotic estimate for the $\Gamma$-dimension of the harmonic space with respect to the tensor times $k$ in the holomorphic line bundle $L^{k}\otimes E$ and the type $(n,q)$ of the differential form, when $L$ is semipositive. In particular, we estimate the $\Gamma$-dimension of the corresponding reduced $L^2$-Dolbeault cohomology group. Essentially, we obtain a local estimate of the pointwise norm of harmonic forms with valued in semipositive line bundles over Hermitian manifolds

Stability of matrix factorization for collaborative filtering

We study the stability vis a vis adversarial noise of matrix factorization algorithm for matrix completion. In particular, our results include: (I) we bound the gap between the solution matrix of the factorization method and the ground truth in terms of root mean square error; (II) we treat the matrix factorization as a subspace fitting problem and analyze the difference between the solution subspace and the ground truth; (III) we analyze the prediction error of individual users based on the subspace stability. We apply these results to the problem of collaborative filtering under manipulator attack, which leads to useful insights and guidelines for collaborative filtering system design.Comment: ICML201

Twisted Gauge Theory Model of Topological Phases in Three Dimensions

We propose an exactly solvable lattice Hamiltonian model of topological phases in $3+1$ dimensions, based on a generic finite group $G$ and a $4$-cocycle $\omega$ over $G$. We show that our model has topologically protected degenerate ground states and obtain the formula of its ground state degeneracy on the $3$-torus. In particular, the ground state spectrum implies the existence of purely three-dimensional looplike quasi-excitations specified by two nontrivial flux indices and one charge index. We also construct other nontrivial topological observables of the model, namely the $SL(3,\mathbb{Z})$ generators as the modular $S$ and $T$ matrices of the ground states, which yield a set of topological quantum numbers classified by $\omega$ and quantities derived from $\omega$. Our model fulfills a Hamiltonian extension of the $3+1$-dimensional Dijkgraaf-Witten topological gauge theory with a gauge group $G$. This work is presented to be accessible for a wide range of physicists and mathematicians.Comment: 37 pages, 9 figures, 4 tables; revised to improve the clarity; references adde