  Sketching is a randomized dimensionality-reduction method that aims to
preserve relevant information in large-scale datasets. Count sketch is a simple
popular sketch which uses a randomized hash function to achieve compression. In
this paper, we propose a novel extension known as Higher-order Count Sketch
(HCS). While count sketch uses a single hash function, HCS uses multiple
(smaller) hash functions for sketching. HCS reshapes the input (vector) data
into a higher-order tensor and employs a tensor product of the random hash
functions to compute the sketch. This results in an exponential saving (with
respect to the order of the tensor) in the memory requirements of the hash
functions, under certain conditions on the input data. Furthermore, when the
input data itself has an underlying structure in the form of various tensor
representations such as the Tucker decomposition, we obtain significant
advantages. We derive efficient (approximate) computation of various tensor
operations such as tensor products and tensor contractions directly on the
sketched data. Thus, HCS is the first sketch to fully exploit the
multi-dimensional nature of higher-order tensors. We apply HCS to tensorized
neural networks where we replace fully connected layers with sketched tensor
operations. We achieve nearly state of the art accuracy with significant
compression on the image classification benchmark.
