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In this paper, we present a new framework to obtain tail inequalities forsums of random matrices. Compared with existing works, our tail inequalitieshave the following characteristics: 1) high feasibility--they can be used tostudy the tail behavior of various matrix functions, e.g., arbitrary matrixnorms, the absolute value of the sum of the sum of the $j$ largest singularvalues (resp. eigenvalues) of complex matrices (resp. Hermitian matrices); and2) independence of matrix dimension --- they do not have the matrix-dimensionterm as a product factor, and thus are suitable to the scenario ofhigh-dimensional or infinite-dimensional random matrices. The price we pay toobtain these advantages is that the convergence rate of the resultinginequalities will become slow when the number of summand random matrices islarge. We also develop the tail inequalities for matrix random series andmatrix martingale difference sequence. We also demonstrate usefulness of ourtail bounds in several fields. In compressed sensing, we employ the resultedtail inequalities to achieve a proof of the restricted isometry property whenthe measurement matrix is the sum of random matrices without any assumption onthe distributions of matrix entries. In probability theory, we derive a newupper bound to the supreme of stochastic processes. In machine learning, weprove new expectation bounds of sums of random matrices matrix and obtainmatrix approximation schemes via random sampling. In quantum information, weshow a new analysis relating to the fractional cover number of quantumhypergraphs. In theoretical computer science, we obtain randomness-efficientsamplers using matrix expander graphs that can be efficiently implemented intime without dependence on matrix dimensions.

On Dimension-free Tail Inequalities for Sums of Random Matrices and Applications