The kinetic Boltzmann equation models macroscopic behavior of gas by averaging individual molecular interactions. It describes the state of gas using seven-dimensional velocity distribution function in three space, three velocity and one temporal variables and is believed to be the most accurate model of non-continuum gas. The flip side of its descriptive power is that a seven-dimensional velocity distribution function is difficult to discretize. As a result, practical solutions of kinetic equations continue to be challenging in three dimensions in complex domains and in applications to complex flows. An additional major difficulty continues to be the evaluation of the multifold integral operator describing the effect of molecular collisions. Low rank tensor approximations emerged recently as a promising approach to reduce effective dimensionality of discrete solutions and accelerate numerical computation of high dimensional problems. Low rank approximations were applied to discretization of the velocity distribution function with some success. At the same time, attempts to compress the collision operator using higher order singular value decomposition usually fail to provide significant savings do the properties of the operator. A possible alternative is the development of reduced order models (ROMs) based on low rank representation of solutions that is specific for the problem at hand. A single evaluation of a ROM for the collision operator requires O(K^3) operation where K is the size of the ROM basis. This becomes prohibitively expensive for K>100 compared to the O(M^2) full rank approaches where M is the total number of discrete velocity points. However, in practice, K