How is linear algebra used in algorithms?

问题:

Several of my peers have mentioned that "linear algebra" is very important when studying algorithms. I've studied a variety of algorithms and taken a few linear algebra courses and I don't see the connection. So how is linear algebra used in algorithms?

For example what interesting things can one with a connectivity matrix for a graph?

回答1:

Three concrete examples:

• Linear algebra is the fundament of modern 3d graphics. This is essentially the same thing that you've learned in school. The data is kept in a 3d space that is projected in a 2d surface, which is what you see on your screen.
• Most search engines are based on linear algebra. The idea is to represent each document as a vector in a hyper space and see how the vector relates to each other in this space. This is used by the lucene project, amongst others. See VSM.
• Some modern compression algorithms such as the one used by the ogg vorbis format is based on linear algebra, or more specifically a method called Vector Quantization.

Basically it comes down to the fact that linear algebra is a very powerful method when dealing with multiple variables, and there's enormous benefits for using this as a theoretical foundation when designing algorithms. In many cases this foundation isn't as appearent as you might think, but that doesn't mean that it isn't there. It's quite possible that you've already implemented algorithms which would have been incredibly hard to derive without linalg.

回答2:

A cryptographer would probably tell you that a grasp of number theory is very important when studying algorithms. And he'd be right--for his particular field. Statistics has its uses too--skip lists, hash tables, etc. The usefulness of graph theory is even more obvious.

There's no inherent link between linear algebra and algorithms; there's an inherent link between mathematics and algorithms.

Linear algebra is a field with many applications, and the algorithms that draw on it therefore have many applications as well. You've not wasted your time studying it.

回答3:

Ha, I can't resist putting this here (even though the other answers are good):

The $25 billion dollar eigenvector.

I'm not going to lie... I never even read the whole thing... maybe I will now :-).

回答4:

I don't know if I'd phrase it as 'linear algebra is very important when studying algorithms". I'd almost put it the other way around. Many, many, many, real world problems end up requiring you to solve a set of linear equations. If you end up having to tackle one of those problems you are going to need to know about some of the many algorithms for dealing with linear equations. Many of those algorithms were developed when computers was a job title, not a machine. Consider gaussian elimination and the various matrix decomposition algorithms for example. There is a lot of very sophisticated theory on how to solve those problems for very large matrices for example.

Most common methods in machine learning end up having an optimization step which requires solving a set of simultaneous equations. If you don't know linear algebra you'll be completely lost.

回答5:

Many signal processing algorithms are based on matrix operations, e.g. Fourier transform, Laplace transform, ...

Optimization problems can often be reduced to solving linear equation systems.

回答6:

Linear algebra is also important in many algorithms in computer algebra, as you might have guessed. For example, if you can reduce a problem to saying that a polynomial is zero, where the coefficients of the polynomial are linear in the variables x1, …, xn, then you can solve for what values of x1, …, xn make the polynomial equal to 0 by equating the coefficient of each x^n term to 0 and solving the linear system. This is called the method of undetermined coefficients, and is used for example in computing partial fraction decompositions or in integrating rational functions.

For the graph theory, the coolest thing about an adjacency matrix is that if you take the nth power of an adjacency Matrix for an unweighted graph (each entry is either 0 or 1), M^n, then each entry i,j will be the number of paths from vertex i to vertex j of length n. And if that isn't just cool, then I don't know what is.

回答7:

All of the answers here are good examples of linear algebra in algorithms.

As a meta answer, I will add that you might be using linear algebra in your algorithms without knowing it. Compilers that optimize with SSE(2) typically vectorize your code by having many data values manipulated in parallel. This is essentially elemental LA.

回答8:

It depends what type of "algorithms".

Some examples:

• Machine-Learning/Statistics algorithms: Linear Regressions (least-squares, ridge, lasso).
• Lossy compression of signals and other processing (face recognition, etc). See Eigenfaces

回答9:

For example what interesting things can one with a connectivity matrix for a graph?

A lot of algebraic properties of the matrix are invariant under permutations of vertices (for example abs(determinant)), so if two graphs are isomorphic, their values will be equal.

This is a source for good heuristics for determining whether two graphs are not isomorphic, since of course equality does not guarantee existance of isomorphism.

Check algebraic graph theory for a lot of other interesting techniques.