Folding grid value vectors of size \(2^L\) into Lth-order tensors of mode size \(2\times \cdots \times 2\), combined with low-rank representation in the tensor train format, has been shown to result in highly efficient approximations for various classes of functions. These include solutions of elliptic PDEs on nonsmooth domains or with oscillatory data. This tensor-structured...
We study the geometric Whitney problem on how a Riemannian manifold (M, g) can be constructed to approximate a metric space \((X,d_X)\). This problem is closely related to manifold interpolation (or manifold reconstruction) where a smooth n-dimensional submanifold \(S\subset {{\mathbb {R}}}^m\), \(m>n\) needs to be constructed to approximate a point cloud in \({{\mathbb {R}}}^m...
We show that computing the interleaving distance between two multi-graded persistence modules is NP-hard. More precisely, we show that deciding whether two modules are 1-interleaved is NP-complete, already for bigraded, interval decomposable modules. Our proof is based on previous work showing that a constrained matrix invertibility problem can be reduced to the interleaving...
We show how the tropical variety of an ideal \(I\unlhd K[x_1,\ldots ,x_n]\) over a field K with non-trivial discrete valuation can always be traced back to the tropical variety of an ideal \(\pi ^{-1}I\unlhd R\llbracket t\rrbracket [x_1,\ldots ,x_n]\) over some dense subring R in its ring of integers. We show that this connection is compatible with the Gröbner polyhedra covering...
The theory of isospectral flows comprises a large class of continuous dynamical systems, particularly integrable systems and Lie–Poisson systems. Their discretization is a classical problem in numerical analysis. Preserving the spectrum in the discrete flow requires the conservation of high order polynomials, which is hard to come by. Existing methods achieving this are...
Recent years have seen a flurry of activities in designing provably efficient nonconvex procedures for solving statistical estimation problems. Due to the highly nonconvex nature of the empirical loss, state-of-the-art procedures often require proper regularization (e.g., trimming, regularized cost, projection) in order to guarantee fast convergence. For vanilla procedures such...
We study how small perturbations of general matrix polynomials may change their elementary divisors and minimal indices by constructing the closure hierarchy (stratification) graphs of matrix polynomials’ orbits and bundles. To solve this problem, we construct the stratification graphs for the first companion Fiedler linearization of matrix polynomials. Recall that the first...
We outline an algorithm to recover the canonical (or, coarsest) stratification of a given finite-dimensional regular CW complex into cohomology manifolds, each of which is a union of cells. The construction proceeds by iteratively localizing the poset of cells about a family of subposets; these subposets are in turn determined by a collection of cosheaves which capture variations...
We study the average condition number for polynomial eigenvalues of collections of matrices drawn from some random matrix ensembles. In particular, we prove that polynomial eigenvalue problems defined by matrices with random Gaussian entries are very well conditioned on the average.
We use techniques from (tracial noncommutative) polynomial optimization to formulate hierarchies of semidefinite programming lower bounds on matrix factorization ranks. In particular, we consider the nonnegative rank, the positive semidefinite rank, and their symmetric analogs: the completely positive rank and the completely positive semidefinite rank. We study convergence...
This paper presents convergence analysis of kernel-based quadrature rules in misspecified settings, focusing on deterministic quadrature in Sobolev spaces. In particular, we deal with misspecified settings where a test integrand is less smooth than a Sobolev RKHS based on which a quadrature rule is constructed. We provide convergence guarantees based on two different assumptions...
Can an ideal I in a polynomial ring $$\Bbbk [\mathbf {x}]$$ over a field be moved by a change of coordinates into a position where it is generated by binomials $$\mathbf {x}^\mathbb A- \lambda \mathbf {x}^\mathbf {b}$$ with $$\lambda \in \Bbbk $$, or by unital binomials (i.e., with $$\lambda = 0$$ or 1)? Can a variety be moved into a position where it is toric? By fibering the G...
In the published article, Figure 5 corresponds to an eigenfunction associated not with the first smallest positive eigenvalue.
The problem of phase retrieval is to determine a signal \(f\in \mathcal {H}\), with \( \mathcal {H}\) a Hilbert space, from intensity measurements \(|F(\omega )|\), where \(F(\omega ):=\langle f, \varphi _\omega \rangle \) are measurements of f with respect to a measurement system \((\varphi _\omega )_{\omega \in \Omega }\subset \mathcal {H}\). Although phase retrieval is always...
We introduce a stochastic model of diffeomorphisms, whose action on a variety of data types descends to stochastic evolution of shapes, images and landmarks. The stochasticity is introduced in the vector field which transports the data in the large deformation diffeomorphic metric mapping framework for shape analysis and image registration. The stochasticity thereby models errors...