Welcome back, my fellow knowledge seekers! In our last session, we delved into the world of solving linear systems of equations, even when the matrix A is not a square matrix. Today, we are going to explore the fascinating concept of least squares regression and its relationship with the singular value decomposition (SVD). So, fasten your seatbelts as we embark on this journey into the realm of optimal solutions!
Unveiling Optimal Solutions
In some cases, we encounter linear systems that are either overdetermined or underdetermined, resulting in either infinitely many solutions or no exact solutions at all. However, fear not, for we can still find an optimal solution using the magical Moore-Penrose pseudoinverse, which is based on the SVD.
In the underdetermined case, where infinite solutions exist, this pseudoinverse solution is indeed the solution. Multiplying matrix A by the pseudoinverse X tilde yields the desired result, B. Moreover, X tilde represents the minimum-norm solution that satisfies the equation.
However, in the overdetermined case, where exact solutions are unattainable, the multiplication of A by X tilde only approximates B. Excitingly, let’s dive into the mechanics of this least squares solution. Brace yourselves as we uncover the source of error in the overdetermined case!
Tracing the Roots of Error
To better understand the error prevalence in the least squares solution, we shall substitute X tilde back into AX = B. Replacing A with USigmaV transpose and X tilde with V Sigma inverse U transpose B, we get the following expression:
AX = (USigmaV transpose)(V Sigma inverse U transpose)BAt first glance, it seems like much of the terms will cancel out, as there are numerous VV transpose, AX, X’, and Sigma elements. However, there’s a catch! If we employ the economy SVD or a truncated SVD, where U only encompasses the first few columns, the matrix UU transpose deviates from the identity matrix. Consequently, UU transpose is no longer equal to the identity matrix, hindering us from obtaining an exact solution.
Now, let’s unravel the fascinating aspect of this derivation. UU transpose B represents the projection of B onto the subspace spanned by the columns of U, which also spans matrix A. This result aligns with our intuition, as B can only be solved by X if it lies within the column space of A. After all, multiplying A by X yields linear combinations of the columns of A.
In the case where A is overcomplete and has more than enough columns to span the entire space, it’s highly likely that B lies within the span of the columns of A. However, in the overdetermined scenario, unless B is specifically crafted to be in the column space of A, the chances are that it will have components that are not spanned by the vectors in A. Therefore, the best we can achieve is to project B onto the column space of A, effectively approximating the component of B that lies within the column space. This, my friends, is the essence of the least squares solution: capturing the best approximation of B by approximating its component in the column space of A.
The Intricacies of Error
As we conclude this enlightening session, it’s crucial to grasp the key takeaway: error arises from an orthogonal projection of B onto the columns of U, which are synonymous with the columns of A. Since we are dealing with a pseudo inverse, error is an inevitable companion in this journey of approximation.
In our next lecture, we will delve into the mathematical conditions that determine the existence (or lack thereof) of solutions for these matrices. This comprehensive understanding will prove invaluable as we navigate the realm of least squares regression and the SVD.
Thank you for joining me today on this fascinating expedition. Stay curious, my friends, and remember to visit Techal for more thrilling adventures in the world of technology. Until next time!
