Related papers: Partition Rank and Partition Lattices
The Laplace expansion expresses the $n \times n$ determinant $\det_n$ as a sum of $n$ products. Do shorter expansions exist? In this paper we: - Fully determine the slice rank decompositions of $\det_n$ (where each product must contain a…
Low-rank tensors appear to be prosperous in many applications. However, the sets of bounded-rank tensors are non-smooth and non-convex algebraic varieties, rendering the low-rank optimization problems to be challenging. To this end, we…
In this paper, we study partitions of finite modules induced by rank support and rank weight. First, we show that partitions induced by rank support are mutually dual with respect to suitable non-degenerate pairings, and hence are…
We introduce subspace rank as a tool for studying ranks of tensors and X-rank more generally. We derive a new upper bound for the rank of a tensor and determine the ranks of partially symmetric tensors in C^2 \otimes C^b \otimes C^b. We…
The Dyson rank of an integer partition is the difference between its largest part and the number of parts it contains. Using Fine-Dyson symmetry, we give formulas for the number of partitions of n with rank larger than n/2, and we prove…
We introduce a framework for benchmarking optimizers according to multiple criteria over various test functions. Based on a recently introduced union-free generic depth function for partial orders/rankings, it fully exploits the ordinal…
We provide a graphical calculus for computing averages of tensor network diagrams with respect to the distribution of random vectors containing independent uniform complex phases. Our method exploits the order structure of the partially…
How to quantify the distance between any two partitions of a finite set is an important issue in statistical classification, whenever different clustering results need to be compared. Developing from the traditional Hamming distance between…
In a series of recent works, we have generalised the consistency results in the stochastic block model literature to the case of uniform and non-uniform hypergraphs. The present paper continues the same line of study, where we focus on…
Structures of multilinear maps are characterized by invariants. In this paper we introduce two invariants, named the isotropy index and the completeness index. These invariants capture the tensorial structure of the kernel of a multilinear…
Refined versions, analytic and combinatorial, are given for classical integer partition theorems. The examples include the Rogers-Ramanujan identities, the Gollnitz-Gordon identities, Euler's odd=distinct theorem, and the Andrews-Gordon…
In the realm of tensor optimization, the low-rank Tucker decomposition is crucial for reducing the number of parameters and for saving storage. We explore the geometry of Tucker tensor varieties -- the set of tensors with bounded Tucker…
Multi-label ranking maps instances to a ranked set of predicted labels from multiple possible classes. The ranking approach for multi-label learning problems received attention for its success in multi-label classification, with one of the…
The main contribution of this note is to establish a framework to extend results of tensor functions over specific field to general field. As a consequence of this framework, we extend the existing work to more general settings: \emph{(1)}…
In this paper, we introduce a natural geometric extension of the partition function. More precisely, we investigate the problem of counting partitions of a rectangle into rectangular blocks with integer sides. Here, two partitions of a…
When factorizing binary matrices, we often have to make a choice between using expensive combinatorial methods that retain the discrete nature of the data and using continuous methods that can be more efficient but destroy the discrete…
We show that the slice rank of the direct sum of two tensors is equal to the sum of their slice ranks. The upper bound is trivial, but the lower bound needs more than a one-line proof, for reasons we explain. This result generalizes the…
We derive exact matrix integral representations for different sums over partitions. The characteristic feature of all obtained matrix models is the presence of logarithmic (or, vice versa, exponential) terms in the potential. Our derivation…
We present a novel method for graph partitioning, based on reinforcement learning and graph convolutional neural networks. Our approach is to recursively partition coarser representations of a given graph. The neural network is implemented…
Graded posets frequently arise throughout combinatorics, where it is natural to try to count the number of elements of a fixed rank. These counting problems are often $\#\textbf{P}$-complete, so we consider approximation algorithms for…