6 The convolution layer 13 6.1 What is a convolution? . . . . . . . . . . . . . . . . . . . . . . . . 13 6.2 Why to convolve? . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 6.3 Convolution as matrix product . . . . . . . . . . . . . . . . . . . 18 6.4 The Kronecker product . . . . . . . . . . . . . . . . . . . . . . . 20 6.5 Backward propagation: update the parameters . . . . . . . . ...

The tensor rank of a tensor t is the smallest number r such that t can be decomposed as a sum of r simple tensors. Let s be a k-tensor and let t be an `-tensor. The tensor product of s and t is a (k + `)-tensor. Tensor rank is sub-multiplicative under the tensor product. We revisit the connection between restrictions and degenerations. A result of our study is that tensor rank is not in general...

Communication speed in a network is related to its diameter. The diameter variability arises from the changes of links from networks. In this paper we study how the diameter of Kronecker product of cycles changes by adding edges.

In this paper we give a combinatorial interpretation for the coefficient of sν in the Kronecker product s(n−p,p) ∗ sλ, where λ = (λ1, . . . , λ`(λ)) ` n, if `(λ) ≥ 2p − 1 or λ1 ≥ 2p − 1; that is, if λ is not a partition inside the 2(p − 1) × 2(p − 1) square. For λ inside the square our combinatorial interpretation provides an upper bound for the coefficients. In general, we are able to combinat...

We explore the combinatorial properties of a particular type of extension monoid product of preinjective Kronecker modules. The considered extension monoid product plays an important role in matrix completion problems. We state theorems which characterize this product in both implicit and explicit ways and we prove that the conditions given in the definition of the generalized majorization are ...

How can we generate realistic graphs? In addition, how can we do so with a mathematically tractable model that makes it feasible to analyze their properties rigorously? Real graphs obey a long list of surprising properties: Heavy tails for the inand out-degree distribution; heavy tails for the eigenvalues and eigenvectors; small diameters; and the recently discovered “Densification Power Law” (...

This paper establishes a straightforward interconnection between the Kronecker canonical form and the special coordinate basis of linear systems. Such an interconnection enables the computation of the Kronecker canonical form, and as a by-product, the Smith form, of the system matrix of general multivariable time-invariant linear systems. The overall procedure involves the transformation of a g...

In this paper we propose and study a technique to impose structural constraints on the output of a neural network, which can reduce amount of computation and number of parameters besides improving prediction accuracy when the output is known to approximately conform to the low-rankness prior. The technique proceeds by replacing the output layer of neural network with the so-called MLM layers, w...

Kronecker algebra is a matrix calculus which allows the generation of thread interleavings from the source-code of a program. Thread interleavings have been shown effective for proving the absence of deadlocks. Because the number of interleavings grows exponentially in the number of threads, deadlock analysis is still a challenging problem. To make the computation of thread interleavings tracta...

Real world graphs have been observed to display a number of surprising properties. These properties include heavy-tails for inand out-degree distributions, small diameters, and a densification law [5]. These features do not arise from the classical Erdos-Renyi random graph model [1]. To address these difficulties, Kronecker Graphs were first introduced in [5] as a new method of generating graph...