Randomized Sketching Matrix for High-Dimensional Regression
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Solution Overview
Problem
Conventional oblivious subspace embeddings (OSEs) are inefficient for statistical learning tasks, particularly in high-dimensional spaces, as they require explicit representation of matrices and are slow for nonlinear kernel-induced spaces, limiting their application in regression and dimensionality reduction.
Innovation Solution
The method introduces a randomized sketching matrix S for processing matrix A and vector b, allowing for fast oblivious subspace embedding and regression in higher dimensional spaces without explicit mapping, using M-sketch construction that reduces the dimensionality of M-estimators and provides constant-factor approximations in polynomial time.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If conventional OSEs are used for high-dimensional statistical learning, then subspace embedding is achieved, but computational speed and efficiency deteriorate
Solution Approach 1:
The patent extracts and processes only the most significant components of high-dimensional data through randomized sketching, selecting a smaller subset of features that capture the essential subspace structure. This extraction approach maintains embedding accuracy while dramatically reducing computational complexity by working with a compressed representation rather than the full high-dimensional space.
Solution Approach 2:
The patent replaces conventional explicit matrix representation and manipulation with randomized sketching techniques that use implicit representations. Instead of directly computing with large explicit matrices, the system uses random projection matrices to transform data into a lower-dimensional sketch, substituting mechanical matrix operations with more efficient randomized algorithms that achieve the same subspace embedding goal.
2Reliability
If explicit matrix representation is used for OSE, then subspace embedding is achieved, but computational complexity increases
Solution Approach 1:
The patent creates a compressed copy or sketch of the original high-dimensional data using randomized projection. Instead of working with the full explicit matrix representation, the system generates a smaller sketch matrix that captures the essential subspace properties. This copying approach preserves the necessary information for subspace embedding while reducing the complexity of subsequent computations.
Solution Approach 2:
The patent changes the dimensional parameters of the data representation by transforming high-dimensional vectors into lower-dimensional sketches through random projection. This parameter transformation maintains the geometric relationships necessary for subspace embedding while operating in a computationally more manageable dimensionality, effectively trading representation dimension for computational efficiency.
3Productivity
If randomized sketching matrix is used for processing, then computational speed improves, but processing accuracy may deteriorate
Solution Approach 1:
The patent applies preliminary randomized sketching transformation to the data before performing regression analysis. By pre-processing the high-dimensional data through random projection into a compressed sketch representation, the system prepares the data in a form that enables faster subsequent computation while preserving the essential regression relationships. This preliminary action maintains accuracy by ensuring the sketch captures the necessary subspace structure.
Solution Approach 2:
The randomized sketching matrix serves as an intermediary that transforms the original high-dimensional data into a compressed representation that retains the essential information needed for regression. This intermediary sketch acts as a bridge between the full-dimensional data and the final regression analysis, enabling computational speedup while maintaining measurement precision through careful design of the sketching process.
Data Source
AI summary
Embodiments of the invention relate to sketching for M-estimators for performing regression. One embodiment includes providing one or more sets of input data. A matrix A and a vector b are generated using the input data. A processor device is used for processing the matrix A and the vector b based on a randomized sketching matrix S. A vector x that minimizes a normalized measure function is determined based on the matrix A and the vector b. A relationship between the input data is determined based on the vector x.