17 research outputs found
On the regularity of special difference divisors
In this note we prove that the difference divisors associated with special
cycles on unitary Rapoport-Zink spaces of signature (1,n-1) in the unramified
case are always regular.Comment: 3 page
Intersections of arithmetic Hirzebruch-Zagier cycles
We establish a close connection between the intersection multiplicity of
three arithmetic Hirzebruch-Zagier cycles and the Fourier coefficients of the
derivative of a certain Siegel-Eisenstein series at its center of symmetry. Our
main result proves a conjecture of Kudla and Rapoport
The supersingular locus of the Shimura variety for GU(1,n-1) over a ramified prime
We analyze the geometry of the supersingular locus of the reduction modulo p
of a Shimura variety associated to a unitary similitude group GU(1,n-1) over Q,
in the case that p is ramified. We define a stratification of this locus and
show that its incidence complex is closely related to a certain Bruhat-Tits
simplicial complex. Each stratum is isomorphic to a Deligne-Lusztig variety
associated to some symplectic group over F_p and some Coxeter element. The
closure of each stratum is a normal projective variety with at most isolated
singularities. The results are analogous to those of Vollaard/Wedhorn in the
case when p is inert.Comment: A few more corrections, to appear in Math. Zeitschrif
Low rank matrix recovery from rank one measurements
We study the recovery of Hermitian low rank matrices from undersampled measurements via nuclear norm minimization. We
consider the particular scenario where the measurements are Frobenius inner
products with random rank-one matrices of the form for some
measurement vectors , i.e., the measurements are given by . The case where the matrix to be recovered
is of rank one reduces to the problem of phaseless estimation (from
measurements, via the PhaseLift approach,
which has been introduced recently. We derive bounds for the number of
measurements that guarantee successful uniform recovery of Hermitian rank
matrices, either for the vectors , , being chosen independently
at random according to a standard Gaussian distribution, or being sampled
independently from an (approximate) complex projective -design with .
In the Gaussian case, we require measurements, while in the case
of -designs we need . Our results are uniform in the
sense that one random choice of the measurement vectors guarantees
recovery of all rank -matrices simultaneously with high probability.
Moreover, we prove robustness of recovery under perturbation of the
measurements by noise. The result for approximate -designs generalizes and
improves a recent bound on phase retrieval due to Gross, Kueng and Krahmer. In
addition, it has applications in quantum state tomography. Our proofs employ
the so-called bowling scheme which is based on recent ideas by Mendelson and
Koltchinskii.Comment: 24 page
On the Arithmetic Fundamental Lemma in the minuscule case
The arithmetic fundamental lemma conjecture of the third author connects the
derivative of an orbital integral on a symmetric space with an intersection
number on a formal moduli space of -divisible groups of Picard type. It
arises in the relative trace formula approach to the arithmetic
Gan-Gross-Prasad conjecture. We prove this conjecture in the minuscule case.Comment: Referee's comments incorporated; in particular, the existence of
frames for using the theory of displays in the proofs of Theorems 9.4 and 9.5
is clarified. To appear in Compositio Mat
Stable low-rank matrix recovery via null space properties
The problem of recovering a matrix of low rank from an incomplete and
possibly noisy set of linear measurements arises in a number of areas. In order
to derive rigorous recovery results, the measurement map is usually modeled
probabilistically. We derive sufficient conditions on the minimal amount of
measurements ensuring recovery via convex optimization. We establish our
results via certain properties of the null space of the measurement map. In the
setting where the measurements are realized as Frobenius inner products with
independent standard Gaussian random matrices we show that
measurements are enough to uniformly and stably recover an
matrix of rank at most . We then significantly generalize this result by
only requiring independent mean-zero, variance one entries with four finite
moments at the cost of replacing by some universal constant. We also study
the case of recovering Hermitian rank- matrices from measurement matrices
proportional to rank-one projectors. For rank-one projective
measurements onto independent standard Gaussian vectors, we show that nuclear
norm minimization uniformly and stably reconstructs Hermitian rank- matrices
with high probability. Next, we partially de-randomize this by establishing an
analogous statement for projectors onto independent elements of a complex
projective 4-designs at the cost of a slightly higher sampling rate . Moreover, if the Hermitian matrix to be recovered is known to be
positive semidefinite, then we show that the nuclear norm minimization approach
may be replaced by minimizing the -norm of the residual subject to the
positive semidefinite constraint. Then no estimate of the noise level is
required a priori. We discuss applications in quantum physics and the phase
retrieval problem.Comment: 26 page