The approximation of one matrix by another of lower rank

By C. Eckart, Gale Young. 1936

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Citations in the corpus

Adafactor: Adaptive Learning Rates with Sublinear Memory Cost (2018, methodology): “This is known to give the optimal projection onto the space of rank-k matrices with respect to the Frobenius norm (Eckart & Young, 1936).”
Linformer: Self-Attention with Linear Complexity (2020, positive): “Based on the results in Theorem 1 and the Eckart–Young–Mirsky Theorem (Eckart & Young, 1936), one can use P low to approximate self-attention (2) with (cid:15) error and O ( nk ) time and space complexity.”
Improving Distributional Similarity with Lessons Learned from Word Embeddings (2015, methodology): “A common method of doing so is truncated Singular Value Decomposition (SVD), which finds the optimal rank d factorization with respect to L2 loss (Eckart and Young, 1936).”
Low-Rank Approximations for Conditional Feedforward Computation in Deep Neural Networks (2013, methodology): “By [6], the matrix A can be approximated using a low rank matrix Âr corresponding to the solution of the constrained optimization of min Âr ‖A− Âr‖F (6)”
Deep Neural Networks for Learning Graph Representations (2016, methodology): “Although the PPMI matrix is a high dimensional matrix, performing dimension reduction using truncated SVD method (Eckart and Young 1936) yields the optimal rank d factorization with respect to L2 loss (Levy and Goldberg 2014).”
Learning Deep Representations for Graph Clustering (2014, methodology): “The Eckart-Young-Mirsky Theorem (Eckart and Young 1936) explains the reconstruction nature of spectral clustering, which is related to the low-rank approximation of a matrix.”
Finding Structure with Randomness: Probabilistic Algorithms for Constructing Approximate Matrix Decompositions (2009, background): “The Eckart– Young theorem [54] shows that ( ∑ j>k σ 2 j ) 1/2 is the minimal Frobenius-norm error when approximating A with a rank-k matrix.”
A Consistent Adjacency Spectral Embedding for Stochastic Blockmodel Graphs (2011, background): “Eckart and Young (1936) provided the following solution to Equation (3).”
A method to construct reduced‐order parameter‐varying models (2017, methodology): “Proof This is a consequence of the Eckart-Young-Mirsky Theorem [28, 29].”
Streaming PCA and Subspace Tracking: The Missing Data Case (2018, background): “While the formulations (2) or (3) are non-convex, both due to the cost function’s non-convexity in U and the non-convex constraint UU = Ik, they admit a well-defined solution, solved by the SVD of Xn, equivalently the EVD of Σn, as was discovered independently by [42] (see [43], [44] for details) an”
Minimum Sum-Squared Residue Co-Clustering of Gene Expression Data (2004, methodology): “We use the fact that the best rank-s approximation to a matrixA, where the approximation error is measured by the Frobenius norm, is given by As [10].”
Data-Driven Sparse Sensor Placement for Reconstruction: Demonstrating the Benefits of Exploiting Known Patterns (2017, methodology): “The SVD is the optimal least squares approximation to the data for a given rank r , as demonstrated by the Eckart-Young theorem [120] X ?”
Fast Convex Optimization Algorithms for Exact Recovery of a Corrupted Low-Rank Matrix (2009, background): “Correctly estimating this subspace is crucial for reducing the dimension of the data and facilitating tasks such as processing, analyzing, compressing, or visualizing the data [12] [7].”
Numerically stable and accurate stochastic simulation approaches for solving dynamic economic models (2011, methodology): “As follows from the theorem of Eckart and Young (1936), X r is the closest rank r approximation of X ∈ R T × n .”