- Series
- Stochastics Seminar
- Time
- Thursday, September 16, 2010 - 3:05pm for 1 hour (actually 50 minutes)
- Location
- Skiles 002
- Speaker
- Vladimir Koltchinskii – School of Mathematics, Georgia Tech
- Organizer
- Yuri Bakhtin
We study a problem of estimation of a large Hermitian nonnegatively definite
matrix S of unit trace based on n independent measurements
Y_j = tr(SX_j ) + Z_j , j = 1, . . . , n,
where {X_j} are i.i.d. Hermitian matrices and {Z_j } are i.i.d. mean
zero random variables independent of {X_j}. Problems of this nature are
of interest in quantum state tomography, where S is an unknown density
matrix of a quantum system. The estimator is based on penalized least
squares method with
complexity penalty defined in terms of von Neumann entropy.
We derive oracle inequalities showing how the estimation error depends on the
accuracy of approximation of the unknown state S by low-rank matrices.
We will discuss these results as well as some of the tools used in their
proofs (such as generic chaining bounds for empirical processes and
noncommutative Bernstein type inequalities).