37 LEAST-SQUARES AND SVD

37.1. Set of solutions

The following theorem provides all the solutions (optimal set) of a least-squares problem.

Theorem: optimal set of ordinary least-squares
 

The optimal set of the OLS problem

    \[ p^* = \min_x \|A x - y\|_2 \]

can be expressed as

    \[ \mathbf{X}^{\text {opt }} = A^{\dagger} y + \mathbf{N}(A), \]

where A^{\dagger} is the pseudo-inverse of A, and A^{\dagger} y is the minimum-norm point in the optimal set. If A is full column rank, the solution is unique, and equal to

    \[ x^* = A^{\dagger} y = (A^T A)^{-1} A^T y. \]

In general, the particular solution A^{\dagger} y is the minimum-norm solution to the least-squares problem.

37.2. Sensitivity analysis

We consider the situation where

    \[ y + \delta y = A x, \]

with

  • A \in \mathbb{R}^{m \times n} the data matrix (known), with A full column rank (hence m \geq n).
  • y \in \mathbb{R}^m is the measurement (known).
  • x \in \mathbb{R}^n is the vector to be estimated (unknown).
  • \delta y \in \mathbb{R}^m is a measurement noise or error (unknown).

We can use OLS to provide an estimate \hat{x}_{\text {OLS }} of X. The idea is to seek the smallest vector \delta y such that the above equation becomes feasible, that is,

    \[ \min_{x, \delta y} \|\delta y\|_2 : y + \delta y = A x. \]

This leads to the OLS problem:

    \[ \min_x \|A x - y\|_2. \]

Since A is full column rank, its SVD can be expressed as

    \[ A = U \begin{pmatrix} \Sigma \\ 0 \end{pmatrix} V^T, \]

where \Sigma=\mathbf{diag}\left(\sigma_1, \ldots, \sigma_m\right) contains the singular values of A, with \sigma_1 \geq \ldots \geq \sigma_m>0.

Since A is full column rank, the solution \hat{x}_{\mathrm{LS}} to the OLS problem is unique, and can be written as a linear function of the measurement vector y:

    \[ \hat{x}_{\mathrm{LS}} = A^{\dagger} y, \]

with A^{\dagger} the pseudo-inverse of A. Again, since A is full column rank,

    \[ A^{\dagger} = (A^T A)^{-1} A^T = V \begin{pmatrix} \Sigma^{-1} & 0 \end{pmatrix} U^T. \]

The OLS formulation provides an estimate \hat{x} of the input x such that the residual error vector Ax-y is minimized in norm. We are interested in analyzing the impact of perturbations in the vector y, on the resulting solution \hat{x}_{LS}. We begin by analyzing the absolute errors in the estimate and then turn to the analysis of relative errors.

Set of possible errors

Let us assume a simple model of potential perturbations: we assume that \delta y belongs to a unit ball: ||\delta y||_2 \leq \alpha, where \alpha>0 is given. We will assume \alpha=1 for simplicity; the analysis is easily extended to any \alpha>0.

We have

    \[ \begin{aligned} \delta x &:= x - \hat{x} \\ &= x - A^{\dagger} y \\ &= x - A^{\dagger}(A x - \delta y) \\ &= (I - A^{\dagger} A) x + A^{\dagger} \delta y \\ &= A^{\dagger} \delta y. \end{aligned} \]

In the above, we have exploited the fact that A^{\dagger} is a left inverse of A, that is, A^{\dagger} A=I_n.

The set of possible errors on the solution \delta x is then given by

    \[ \mathbf{E} = \left\{ A^{\dagger} \delta y : \|\delta y\|_2 \leq 1 \right\}, \]

which is an ellipsoid centered at zero, with principal axes given by the singular values of A^{\dagger}. This ellipsoid can be interpreted as an ellipsoid of confidence for the estimate \hat{x}, with size and shape determined by the matrix A^{\dagger}(A^{\dagger})^T.

We can draw several conclusions from this analysis:

  • The largest absolute error in the solution that can result from a unit-norm, additive perturbation on y is of the order of 1 / \sigma_m, where \sigma_m is the smallest singular value of A.
  • The largest relative error is \sigma_1 / \sigma_m, the condition number of A.

37.3. BLUE property

We now return to the case of an OLS with full column rank matrix A.

Unbiased linear estimators

Consider the family of linear estimators, which are of the form

    \[ \hat{x} = B y, \]

where B \in \mathbb{R}^{n \times m}. To this estimator, we associate the error

    \[ \begin{aligned} \delta x &= x - \hat{x} \\ &= x - B y \\ &= x - B(A x - \delta y) \\ &= (I - B A) x + B \delta y. \end{aligned} \]

We say that the estimator (as determined by matrix B) is unbiased if the first term is zero:

    \[ B A = I. \]

Unbiased estimators only exist when the above equation is feasible, that is, A has a left inverse. This is equivalent to our condition that A be full column rank. Since A^{\dagger} is a left-inverse of A, the OLS estimator is a particular case of an unbiased linear estimator.

Best unbiased linear estimator

The above analysis leads to the following question: which is the best unbiased linear estimator? One way to formulate this problem is to assume that the perturbation vector \delta y is bounded in some way, and try to minimize the possible impact of such bounded errors on the solution.

Let us assume that \delta y belongs to a unit ball: \|\delta y\|_2 \leq 1. The set of resulting errors on the solution \delta x is then

    \[ \mathbf{E} = \left\{ B \delta y : \|\delta y\|_2 \leq 1 \right\}, \]

which is an ellipsoid centered at zero, with principal axes given by the singular values of B. This ellipsoid can be interpreted as an ellipsoid of confidence for the estimate \hat{x}, with size and shape determined by the matrix BB^T.

It can be shown that the OLS estimator is optimal in the sense that it provides the ‘‘smallest’’ ellipsoid of confidence among all unbiased linear estimators. Specifically:

    \[ B B^T \succeq A^{\dagger} (A^{\dagger})^T. \]

This optimality of the LS estimator is referred to as the BLUE (Best Linear Unbiased Estimator) property.

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