Mathematics

Questions tagged [matrix-completion]

Matrix completion is the task of filling in the missing entries of a partially observed matrix.

39 questions
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Recover primal solution from dual for matrix completion

Consider the following primal/dual SDPs $$ \min\limits_X \; \lVert X \rVert_* : \mathcal{A}(X) = b \qquad \max\limits_z \; b^T z : \lVert \mathcal{A}^*(z) \rVert \leq 1 $$ where $\lVert X \rVert_* = \sum_{i=1}^{\text{rank}(X)} \sigma_i(X)$ is the…
steve
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6
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Positive definite completion of a matrix

Suppose we have a real, symmetric matrix $A(x_1,x_2,x_3)$ given by \begin{pmatrix} a_{1,1} & a_{1,2} & x_1 & x_2 \\ a_{2,1} & a_{2,2} & a_{2,3} & x_3 \\ x_1 & a_{3,2} & a_{3,3} & a_{3,4} \\ x_2 & x_3 & a_{3,4} & a_{4,4} \end{pmatrix} We would like…
BharatRam
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Finding matrix with lowest possible rank

Find the values of $x$ for which the matrix \begin{bmatrix} x & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & x \\ \end{bmatrix} has the lowest rank. Since the rank is the dimension of the row space of $A$ , I should find $x$ for which the dimension of the row…
zzxxx123
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How do I complete a matrix so that its columns are orthogonal?

I'm not sure how to start on these types of problems. I know that the dot product of the columns have to be zero between each pair. Do I set up a system of linear equations for this? If so, what would it even look like?
Samurai Bale
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How does minimizing the rank of a matrix help us impute missing values in it?

I am not really a math guru myself, but I know that many estimation or approximation problems can be reformulated as minimizing the rank of a matrix. Although that is really hard, we can try to minimize the nuclear norm instead, which turns out to…
MLearner
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How to convert the matrix completion problem to the standard SDP form?

Given the matrix completion problem defined below. \begin{equation} \begin{array}{ll} \text{minimize }{X \in \mathbb{R}^{m \times n}} & \sum_{(i,j) \in \Omega} ( X_{ij} - Z_{ij} )^2 + \lambda \text{trace}(\sqrt[]{X^TX}) , \end{array} \end{equation}…
good2know
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Question on Candès-Tao's near-optimal matrix completion theorem

I have a question about the key result from this paper : Candès, Emmanuel J.; Tao, Terence , The power of convex relaxation: near-optimal matrix completion , IEEE Trans. Inf. Theory 56, No. 5, 2053-2080 (2010). ZBL1366.15021 . The question…
Audrey
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Bound determined by Coherence of matrix using SVD for Matrix Completion

So I'm referring to the paper Exact Matrix Completion via Convex Optimization: https://arxiv.org/pdf/0805.4471.pdf If you look on page 6 of this paper, The coherence of a matrix $\mu(U) = \frac{n}{r} \max_{1 \le i \le n} \|P_Ue_i\|_2^2$ where $i$ is…
cashinhand
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"Shadow prices" interpretation of the dual certificate of nuclear norm optimization

In nuclear norm based matrix completion, there exists a low rank matrix $M$ of which only the indicies in $\Omega$ are sampled. Candes and Recht (2008) recover the original matrix using following optimization problem. $$ \begin{align} \text{minimize…
The_Anomaly
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2
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1 answer

Low-rank matrix satisfying linear constraints linear mapping

Let $X \in \mathbb{R}^{m \times n}$ be a matrix that is assumed to be low rank. According to, Recht, Benjamin; Fazel, Maryam; Parrilo, Pablo A. , Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization , SIAM Rev.…
P. Kell
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Completion of $2 \times 2$ positive semidefinite rank-$1$ partial matrix

This question is related to one property of rank-$1$, positive semidefinite matrices. Would be very useful in SDP problems (which is where I found it). Consider a $3 \times 3$ positive semidefinite rank-$1$ matrix $A = xx^H$ $$ $$ \begin{bmatrix}…
Ben Wu
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Phase retrieval via SDP (semidefinite program) of 2D test image (Matrix completion)

In SDP based phase retrieval we have intensity measurement (abs(FFT2(x))^2) of the form A(xo) = |ak,x|^2 =b^2, phase retrieval is then find x that obeys A(xo)=b The quadratic measurements can be lifted and interpreted as linear about rank 1 matrix X…
Anonymous
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Completing a unitary matrix given a column

I am given a unit vector $e=1/\sqrt{n}(1,1,\ldots,1)'$ and the problem is to construct an $n \times n$ (real) unitary matrix $U$ which will contain $e$ as the last column. I understand that there are infinitely many such $U$ ($n>2$). I wonder if…
Tapu
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SDP formulation of noisy low rank matrix completion (2)

Thank you, Michael, for answering my previous question . It seems that I overlooked the problems in my initial question. I didn't recognize the SDP (in either primal or dual form) even for the 2-norm constrained problem. Now, I have a better…
tngo
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Incoherence property in Matrix Completion

I'm not a math major and I find the statement below confusing (from a paper by Candes and Recht on Matrix Completion). Can someone clarify this? I'm sure it's painfully simple/obvious. "For instance, consider the rank-2 symmetric matrix M given by…
val
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