Introduces P-tensors for generalized higher-order message passing in graph neural networks.
problem Expanding the expressive power of graph neural networks through higher-order structures.
method Introduces P-tensors to define the most general form of permutation equivariant message passing.
result Achieves state-of-the-art performance on molecular datasets.
Estimates natural parameters of p-tensor Ising models efficiently.
problem Estimating natural parameters of p-tensor Ising models from a single sample.
method Maximum pseudo-likelihood (MPL) method.
result MPL estimate is sqrt(N)-consistent under various conditions.
Upper bound found for divergence-free Killing 2-tensors on manifolds.
problem Bounding the space of divergence-free symmetric Killing 2-tensors.
method Witten deformation and Morse function analysis.
result Explicit calculation of dimension for p=2. We establish a new algebraic characterization of sectional curvature bounds sec≥k and sec≤k using only curvature terms in the Weitzenböck formulae for symmetric p-tensors. By introducing a symmetric analogue of the Kulkarni-Nomizu product, we provide a simple formula for such curvature terms. We also gi…
The largest class of Riemannian almost product manifolds, which is closed with respect to the group of the conformal transformations of the Riemannian metric, is the class of the conformal Riemannian P-manifolds. This class is an analogue of the class of the conformal Kähler manifolds in almost Hermitian geometry. The …
It is known that a compact symplectic manifold endowed with a prequantum line bundle can be embedded in the projective space generated by the eigensections of low energy of the Bochner Laplacian acting on high p-tensor powers of the prequantum line bundle. We show that the Fubini-Study forms induced by these embeddin…
More than forty years ago J. H. Samson has defined the Laplacian Δsym acting on the space of symmetric covariant p-tensors on an n-dimensional Riemannian manifold (M,g). This operator is an analogue of the well known Hodge-de Rham Laplacian Δ which acts on the space of exterior differential p-forms ($1 …
Efficient algorithm for tensor PCA with improved time complexity.
problem Tensor PCA problem with signal-to-noise ratio constraint.
method Counting specific weighted hypergraphs to solve tensor PCA.
result Algorithm runs in time nC+o(1) for λn−4p signal-to-noise ratio. Neural networks learn faster with correlated latent variables.
problem Efficiently learning from higher-order correlations in neural networks.
method Analytical derivation and simulations of two-layer neural networks.
result Correlations between latent variables speed up learning from higher-order correlations.
Geometrically proves Zabrodin-Wiegmann conjecture for integer QH states.
problem Proving a geometric version of Zabrodin-Wiegmann conjecture for integer Quantum Hall states.
method Using Riemann surfaces, canonical sections, and asymptotic expansions, the authors construct a canonical element in cohomology and relate its norm to the partition function.
result The constant term of the asymptotic expansion of the partition function matches a geometric version of Zabrodin-Wiegmann's prediction.
SGD recovers multiple signal vectors in noisy tensor PCA.
problem Estimating multiple signal vectors from noisy tensor observations.
method Online stochastic gradient descent (SGD) in high dimensions with detailed analysis of correlations.
result Sequential elimination of correlations allows recovery of all spikes from Np−2 samples. New algorithms detect categorical structures in high-dimensional data.
problem Detecting categorical structures in high-dimensional data.
method Low coordinate degree functions (LCDF) applied to categorical and stochastic block models.
result Unified analysis of LCDF performance for various SBMs and tight lower bounds.