Kronecker Product Matrix Decompression for Data Storage
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Solution Overview
Problem
Existing compression methods based on Singular Value Decomposition (SVD) face challenges in achieving desirable compression ratios due to the need to store non-integer singular values and vectors, leading to increased memory usage and inefficiencies, especially when compared to methods like JPEG.
Innovation Solution
The Bourouihiya Schmidt Decomposition (BSD) method decomposes matrices into Kronecker products of smaller matrices, allowing for lossless and lossy compression by selecting essential entries and using elementary operations, which reduces storage requirements and improves compression efficiency.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Quantity of substance
If SVD-based compression is used, then compression is achieved, but memory usage increases due to storing non-integer singular values and vectors
Solution Approach 1:
The patent segments the matrix decomposition into Kronecker products of smaller matrices, where each component matrix can be stored separately. This segmentation allows for more efficient storage compared to storing full singular value decompositions, as the Kronecker structure enables compact representation of large matrices through smaller factor matrices.
Solution Approach 2:
The patent changes the mathematical parameters from standard SVD components (singular values and vectors) to Kronecker product components. This parameter transformation allows the compressed representation to use integer or simplified values instead of complex non-integer singular values, reducing memory requirements while maintaining compression effectiveness.
2Reliability
If SVD compression stores singular values and vectors, then reconstruction is possible, but storage space per pixel increases significantly
Solution Approach 1:
The patent transitions from storing compression parameters in the original matrix dimension to storing them in a factored Kronecker dimension. By representing an mn × pq matrix as Kronecker products of smaller matrices, the storage requirement shifts from O(mn×pq) to O((m×p)+(n×q)), effectively reducing the dimensional burden and storage space per pixel while preserving reconstruction capability.
3Productivity
If BSD method is used, then compression efficiency improves, but computational complexity increases due to matrix decomposition operations
Solution Approach 1:
The BSD method segments the complex matrix decomposition task into finding Kronecker product components, which can be computed through systematic algorithms. This segmentation breaks down the computational complexity into manageable steps compared to traditional SVD, particularly when dealing with structured matrices that admit Kronecker decompositions.
Solution Approach 2:
The patent uses copying techniques where the decomposed Kronecker components are stored and reused for multiple operations. Once the matrix is decomposed into Kronecker factors, these factors can be copied and used for compression, encryption, and reconstruction without repeating the full decomposition process, thereby improving compression efficiency after the initial computational investment.
Data Source
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AI summary
Digital files are compressed using a process including Schmidt decompositions of matrices using an algorithm, termed 'BSD' herein, which is based on an algebraic method generalizing QR decomposition. Software analyzes an input file and initially identifies a matrix M, with entries within a predefined set of integers, within the file. Next, essential entries are defined, extracted from M, that contain sufficient information to recover Musing BSD. The compressed file includes the essential entries and their positions within M. To achieve an encryption process, software encrypts the pattern matrix that includes the positions of the essential entries of M. To achieve a lossy compression, software identifies essential entries that contain sufficient information to recover an approximation to M for which the quality is determined by an error threshold. For a more efficient lossy compression, software uses singular value decomposition, BSD, and other signal processing of M.