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Math Help - Optimal Approximation

  1. #1
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    Optimal Approximation

    Hello,

    1) Let v \in \mathbb{R}^d be a unit vector. What is the meaning of the linear mapping A:\mathbb{R}^d \rightarrow \mathbb{R}^d, defined as

    A u = vv^T u

    2) Which linear subspace is the best approximation to the set of n points in the sense that the Euclidean distance to the subspace is minimized:

    v^{*} = argmin_{v \in \mathbb{R}^d , \Vert v \Vert = 1} (\frac{1}{n} \sum_{i=1}^d \Vert X_i - (vv^T)X_i \Vert^2)
    Thank you!
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  2. #2
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    Opalg's Avatar
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    1) The mapping A is the orthogonal projection onto the one-dimensional subspace spanned by the unit vector v.

    2) So the unit vector v that minimises the expression \frac1n\sum_{i=1}^d\|X_i - vv^{\textsc t}X_i\|^2 spans a line (=one-dimensional subspace) that is the best approximation to the set of d points X_1,\ldots,X_d in the sense that the mean of the squares of the Euclidean distances from the points to the line is minimised.
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  3. #3
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    Thanks! I am now trying to find an expression for such a v...
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