  In this paper, we present a new framework to obtain tail inequalities for
sums of random matrices. Compared with existing works, our tail inequalities
have the following characteristics: 1) high feasibility--they can be used to
study the tail behavior of various matrix functions, e.g., arbitrary matrix
norms, the absolute value of the sum of the sum of the $j$ largest singular
values (resp. eigenvalues) of complex matrices (resp. Hermitian matrices); and
2) independence of matrix dimension --- they do not have the matrix-dimension
term as a product factor, and thus are suitable to the scenario of
high-dimensional or infinite-dimensional random matrices. The price we pay to
obtain these advantages is that the convergence rate of the resulting
inequalities will become slow when the number of summand random matrices is
large. We also develop the tail inequalities for matrix random series and
matrix martingale difference sequence. We also demonstrate usefulness of our
tail bounds in several fields. In compressed sensing, we employ the resulted
tail inequalities to achieve a proof of the restricted isometry property when
the measurement matrix is the sum of random matrices without any assumption on
the distributions of matrix entries. In probability theory, we derive a new
upper bound to the supreme of stochastic processes. In machine learning, we
prove new expectation bounds of sums of random matrices matrix and obtain
matrix approximation schemes via random sampling. In quantum information, we
show a new analysis relating to the fractional cover number of quantum
hypergraphs. In theoretical computer science, we obtain randomness-efficient
samplers using matrix expander graphs that can be efficiently implemented in
time without dependence on matrix dimensions.
