How to minimize a simple quadratic function of multiple variables ?

In summary, the conversation is about minimizing a function with multiple variables, specifically approximating a matrix using the outer product of two vectors. The goal is to determine a vector y and a vector x that minimizes the sum of squared differences between the outer product and the given matrix. While it is possible to solve this using gradient descent, the speaker is looking for an analytical solution but is unsure of how to approach the problem due to the complexity of variables. They compare it to linear regression and suggest looking into that method for guidance.
  • #1
darwid
2
0
Hi everybody,

I'm trying to minimize a function with multiple variables. My goal is to approximate on the L2 norm a matrix by the outer product of 2 vectors (or is it called tensor product ?).

So I have to determine a vector y = (y1,...,yn) and a vector x = (x1,...,xm) such that their outer product approximates a given matrix A = (ai,j), i=1..n, j=1..m

What I want to minimize is thus:
s = [tex]\sum[/tex](yixj-ai,j)2

Obviously I can solve this using a gradient descent and it works.

But what I'm looking for is an analytical solution. The formulation looks simple so I expect there must be some analytical way of solving this, it's just that I don't really know how to approach this problem due to the many variables.

--
Darwid
 
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  • #2

1. What is a simple quadratic function of multiple variables?

A simple quadratic function of multiple variables is a mathematical expression that includes multiple variables raised to the second power (or squared) and has a constant term. It can be written in the form of f(x,y) = ax² + by² + cxy + dx + ey + f, where a, b, c, d, e, and f are constants.

2. How do you minimize a simple quadratic function of multiple variables?

In order to minimize a simple quadratic function of multiple variables, you need to find the values of the variables that result in the lowest possible output of the function. This can be done by using calculus techniques, such as taking partial derivatives and setting them equal to zero, or by using optimization algorithms.

3. What is the purpose of minimizing a simple quadratic function of multiple variables?

The purpose of minimizing a simple quadratic function of multiple variables is to find the optimal values for the variables that will result in the lowest output of the function. This is useful in many fields, such as economics, engineering, and physics, where minimizing a cost or maximizing a benefit is desired.

4. Can a simple quadratic function of multiple variables have more than one minimum value?

Yes, a simple quadratic function of multiple variables can have more than one minimum value. This happens when the function has multiple local minima, which are points where the output of the function is lower than the output at any nearby points. In some cases, there may also be a global minimum, which is the absolute lowest value of the function.

5. Are there any techniques for minimizing a simple quadratic function of multiple variables without using calculus?

Yes, there are some techniques for minimizing a simple quadratic function of multiple variables without using calculus. These include gradient descent, which is an iterative algorithm that uses the gradient of the function to find the minimum, and the simplex method, which is a linear programming technique that can also be applied to quadratic functions.

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