한 줄 요약
low rank 란 행렬를 SVD 했을 때, singular value의 개수가 행렬의 행이나 열의 개수보다 작아진 경우를 말한다!
ChatGPT3.5
what is the definition of low rank matrix
A low-rank matrix is a matrix whose rank is significantly smaller than one of its dimensions (either the number of rows or the number of columns). In other words, a matrix is considered low-rank if it can be well approximated by a matrix of lower rank. The rank of a matrix is the maximum number of linearly independent rows or columns it contains.
Formally, for an (m \times n) matrix (A), where (m) is the number of rows and (n) is the number of columns:
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If (m \leq n), the matrix (A) is said to be low-rank if (\text{rank}(A) \leq m).
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If (n \leq m), the matrix (A) is said to be low-rank if (\text{rank}(A) \leq n).
In practical terms, low-rank matrices often exhibit some structure or redundancy, and they can be efficiently approximated using a smaller number of parameters than a full-rank matrix. This property is exploited in various applications, such as data compression, signal processing, collaborative filtering, and matrix completion.
Low-rank matrices are also important in the context of singular value decomposition (SVD), where a matrix (A) can be decomposed into three matrices (U), (\Sigma), and (V^T) such that:
[ A = U \Sigma V^T ]
Here, (\Sigma) is a diagonal matrix containing the singular values of (A). The rank of (A) is equal to the number of non-zero singular values in (\Sigma), and a low-rank approximation can be obtained by retaining only the top (k) singular values and their corresponding vectors in (U) and (V).
