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Sparse Approximate Inverse for Enhanced Scalability in Recommender Systems
Spišák, Martin ; Peška, Ladislav (advisor) ; Vančura, Vojtěch (referee)
In theory, the linear autoencoder EASE is one of the most capable collaborative filtering recommenders for large item domains with sparse user-item feedback. However, the model's weights are determined by the inverse of a matrix of dimension equal to the item set size. This inverse matrix is generally dense, and for large item sets, the computed weight matrix might be too large to store in memory during inference. Consequently, scaling the model beyond tens of thousands of items quickly becomes very expensive. We propose a modification of EASE called SANSA to alleviate the issue. SANSA approximates the weights of EASE with prescribed density via an end-to-end sparse training procedure. To find a method capable of computing the sparse approximation efficiently, we investigate approaches for constructing sparse approximate inverse precon- ditioners. We select a method fitting for very large SPD problems with general sparsity patterns. The training procedure is robust and finds a good approximation of EASE even on datasets with dense item relations. Moreover, as the number of items in datasets grows, SANSA achieves unparalleled efficiency, even compared to EASE's previous state- of-the-art modification focused on scalability. Consequently, SANSA effortlessly scales the concept of EASE to millions of items. 1
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Deriving a pseudomanifold of dimension 3 from nonassociative triples
Spišák, Martin ; Drápal, Aleš (advisor) ; Patáková, Zuzana (referee)
The non-associative properties of quasigroups are useful in cryptography. A. Drápal and I. M. Wanless have recently analyzed the existence of a max- imally non-associative quasigroup of order n in their work, but there remain orders n for which the existence is not known. This thesis is an introduction to a new method of tackling the problem. After presenting the most recent results and hinting at a possible crypto- graphic application, the thesis proposes the construction of a 3-dimensional abstract simplicial complex from non-associative triples of a finite quasigroup. It shows that the complex forms of a union of closed orientable pseudomani- folds of dimension 3. For orders up to 6, we independently verify the findings of Ježek and Kepka regarding the associativity spectrum of n and classify possible decompositions of the non-associativity complexes into strongly con- nected components by analyzing their dual graphs. The main result of the thesis performs the first step towards resolving the singularities in the complex. We show that links of vertices in the complex have solvable singularities, enabling us to normalize the links of vertices algorithmically. Lastly, we illustrate the types of vertex neighborhoods on examples of small quasigroups by calculating the genera of their components. 1
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