Electronics&Communication · Mathematic

l0-Norm, l1-Norm, l2-Norm, … , l-infinity Norm

I’m working on things related to norm a lot lately and it is time to talk about it. In this post we are going to discuss about a whole family of norm.

What is a norm?

Mathematically a norm is a total size or length of all vectors in a vector space  or matrices. For simplicity, we can say that the higher the norm is, the bigger the (value in) matrix or vector is. Norm may come in many forms and many names, including these popular name: Euclidean distance, Mean-squared Error, etc.

Most of the time you will see the norm appears in a equation like this:

\left \| x \right \| where x can be a vector or a matrix.

For example, a Euclidean norm of a vector a = \begin{bmatrix}  3  \\  -2  \\  1  \end{bmatrix} is \left \| a \right \|_2=\sqrt{3^2+(-2)^2+1^2}=3.742 which is the size of vector a

The above example shows how to compute a Euclidean norm, or formally called an l_2-norm. There are many other types of norm that beyond our explanation here, actually for every single real number, there is a norm correspond to it (Notice the emphasised word real number, that means it not limited to only integer.)

Formally the l_p-norm of x is defined as:

\left \| x \right \|_p = \sqrt[p]{\sum_{i}\left | x_i \right |^p}  where p \epsilon \mathbb{R}

That’s it! A p-th-root of a summation of all elements to the p-th power is what we call a norm.

The interesting point is even though every l_p-norm is all look  very similar to each other, their mathematical properties are very different and thus their application are dramatically different too. Hereby we are going to look into some of these norms in details.

l0-norm 

The first norm we are going to discuss is a l_0-norm. By definition, l_0-norm of x is

\left \| x \right \|_0 = \sqrt[0]{\sum_{i}x_i^0}

Strictly speaking, l_0-norm is not actually a norm. It is a cardinality function which has its definition in the form of l_p-norm, though many people call it a norm. It is a bit tricky to work with because there is a presence of zeroth-power and zeroth-root in it. Obviously any x>0 will become one, but the problems of the definition of zeroth-power and especially zeroth-root is messing things around here. So in reality, most mathematicians and engineers use this definition of l_0-norm instead:

\left \| x \right \|_0 = \#(i | x_i \neq 0)

that is a total number of non-zero elements in a vector.

Because it is a number of non-zero element, there is so many applications that use l_0-norm. Lately it is even more in focus because of the rise of the Compressive Sensing scheme, which is try to find the sparsest solution of the under-determined linear system. The sparsest solution means the solution which has fewest non-zero entries, i.e. the lowest l_0-norm. This problem is usually regarding as a optimisation problem of l_0-norm or l_0-optimisation.

l0-optimisation

Many application, including Compressive Sensing, try to minimise the l_0-norm of a vector corresponding to some constraints, hence called “l_0-minimisation”. A standard minimisation problem is formulated as:

min \left \| x \right \|_0 subject to Ax = b

However, doing so is not an easy task. Because the lack of l_0-norm’s mathematical representation, l_0-minimisation is regarded by computer scientist as an NP-hard problem, simply says that it’s too complex and almost impossible to solve.

In many case, l_0-minimisation problem is relaxed to be higher-order norm problem such as l_1-minimisation and l_2-minimisation.

l1-norm

Following the definition of norm, l_1-norm of x is defined as

\left \| x \right \|_1 = \sum_{i} \left | x_i \right |

This norm is quite common among the norm family. It has many name and many forms among various fields, namely Manhattan norm is it’s nickname. If the l_1-norm is computed for a difference between two vectors or matrices, that is

SAD(x_1,x_2) = \left \| x_1-x_2 \right \|_1 = \sum \left | x_{1_i}-x_{2_i} \right |

it is called Sum of Absolute Difference (SAD) among computer vision scientists.

In more general case of signal difference measurement, it may be scaled to a unit vector by:

MAE(x_1,x_2) = \frac{1}{n} \left \| x_1-x_2 \right \|_1 = \frac {1} {n} \sum \left | x_{1_i} - x_{2_i} \right | where n is a size of x.

which is known as Mean-Absolute Error (MAE).

l2-norm

The most popular of all norm is the l_2-norm. It is used in almost every field of engineering and science as a whole. Following the basic definition, l_2-norm is defined as

\left \| x \right \|_2 = \sqrt{\sum_{i}x_i^2}

l_2-norm is well known as a Euclidean norm, which is used as a standard quantity for measuring a vector difference. As in l_1-norm, if the Euclidean norm is computed for a vector difference, it is known as a Euclidean distance:

\left \| x_1-x_2 \right \|_2 = \sqrt{\sum_i (x_{1_i}-x_{2_i})^2}

or in its squared form, known as a Sum of Squared Difference (SSD) among Computer Vision scientists:

SSD(x_1,x_2) = \left \| x_1-x_2 \right \|_2^2 = \sum_i (x_{1_i}-x_{2_i})^2

It’s most well known application in the signal processing field is the Mean-Squared Error (MSE) measurement, which is used to compute a similarity, a quality, or a  correlation between two signals. MSE is

MSE(x_1,x_2) = \frac{1}{n} \left \| x_1-x_2 \right \|_2^2 = \frac{1}{n} \sum_i (x_{1_i}-x_{2_i})^2

As previously discussed in l_0-optimisation section, because of many issues from both a computational view and a mathematical view, many l_0-optimisation problems relax themselves to become l_1– and l_2-optimisation instead. Because of this, we will now discuss about the optimisation of l_2.

l2-optimisation

As in l_0-optimisation case, the problem of minimising l_2-norm is formulated by

min \left \| x \right \|_2 subject to Ax = b

Assume that the constraint matrix A has full rank, this problem is now a underdertermined system which has infinite solutions. The goal in this case is to draw out the best solution, i.e. has lowest l_2-norm, from these infinitely many solutions. This could be a very tedious work if it was to be computed directly. Luckily it is a mathematical trick that can help us a lot in this work.

By using a trick of Lagrange multipliers, we can then define a Lagrangian

\mathfrak{L}(\boldsymbol{x}) = \left \| \boldsymbol{x} \right \|_2^2+\lambda^{T}(\boldsymbol{Ax}-\boldsymbol{b})

where \lambda is the introduced Lagrange multipliers. Take derivative of this equation equal to zero to find a optimal solution and get

\hat{\boldsymbol{x}}_{opt} = -\frac{1}{2} \boldsymbol{A}^{T} \lambda

plug this solution into the constraint to get

\boldsymbol{A}\hat{\boldsymbol{x}}_{opt} = -\frac{1}{2}\boldsymbol{AA}^{T}\lambda=\boldsymbol{b}

\lambda=-2(\boldsymbol{AA}^{T})^{-1}\boldsymbol{b}

and finally

\hat{\boldsymbol{x}}_{opt}=\boldsymbol{A}^{T} (\boldsymbol{AA}^{T})^{-1} \boldsymbol{b}=\boldsymbol{A}^{+} \boldsymbol{b}

By using this equation, we can now instantly compute an optimal solution of the l_2-optimisation problem. This equation is well known as the Moore-Penrose Pseudoinverse and the problem itself is usually known as Least Square problem, Least Square regression, or Least Square optimisation.

However, even though the solution of Least Square method is easy to compute, it’s not necessary be the best solution. Because of the smooth nature of l_2-norm itself,  it is hard to find a single, best solution for the problem.

In contrary, the l_1-optimisation can provide much better result than this solution.

l1-optimisation

As usual, the l_1-minimisation problem is formulated as

min \left \| x \right \|_1 subject to Ax = b

Because the nature of l_1-norm is not smooth as in the l_2-norm case, the solution of this problem is much better and more unique than the l_2-optimisation.

However, even though the problem of l_1-minimisation has almost the same form as the l_2-minimisation, it’s much harder to solve. Because this problem doesn’t have a smooth function, the trick we used to solve l_2-problem is no longer valid.  The only way left to find its solution is to search for it directly. Searching for the solution means that we have to compute every single possible solution to find the best one from the pool of “infinitely many” possible solutions.

Since there is no easy way to find the solution for this problem mathematically, the usefulness of l_1-optimisation is very limited for decades. Until recently, the advancement of computer with high computational power allows us to “sweep” through all the solutions. By using many helpful algorithms, namely the Convex Optimisation algorithm such as linear programming, or non-linear programming, etc. it’s now possible to find the best solution to this  question. Many applications that rely on l_1-optimisation, including the Compressive Sensing, are now possible.

There are many toolboxes  for l_1-optimisation available nowadays.  These toolboxes usually use different approaches and/or algorithms to solve the same question. The example of these toolboxes are l1-magic, SparseLab, ISAL1,

Now that we have discussed many members of norm family, starting from l_0-norm, l_1-norm, and l_2-norm. It’s time to move on to the next one. As we discussed in the very beginning that there can be any l-whatever norm following the same basic definition of norm, it’s going to take a lot of time to talk about all of them. Fortunately, apart from l_0-, l_1– , and l_2-norm, the rest of them usually uncommon and therefore don’t have so many interesting things to look at. So we’re going to look at the extreme case of norm which is a l_{\infty}-norm (l-infinity norm).

l-infinity norm

As always, the definition for l_{\infty}-norm is

\left \| x \right \|_{\infty} = \sqrt[\infty]{\sum_i x_i^{\infty}}

Now this definition looks tricky again, but actually it is quite strait forward. Consider the vector \boldsymbol{x}, let’s say if x_j is the highest entry in the vector  \boldsymbol{x}, by the property of the infinity itself, we can say that

x_j^{\infty}\gg x_i^{\infty} \forall i \neq j

 then

\sum_i x_i^{\infty} = x_j^{\infty}

then

\left \| x \right \|_{\infty} = \sqrt[\infty]{\sum_i x_i^{\infty}} = \sqrt[\infty]{x_j^{\infty}} = \left | x_j \right |

Now we can simply say that the l_{\infty}-norm is

\left \| x \right \|_{\infty} = max(\left | x_i \right |)

that is the maximum entries’ magnitude of that vector. That surely demystified the meaning of l_{\infty}-norm

Now we have discussed the whole family of norm from l_0 to l_{\infty}, I hope that this discussion would help understanding the meaning of norm, its mathematical properties, and its real-world implication.

Reference and further reading:

Mathematical Norm – wikipedia 

Mathematical Norm – MathWorld

Michael Elad – “Sparse and Redundant Representations : From Theory to Applications in Signal and Image Processing” , Springer, 2010.

 Linear Programming – MathWorld

Compressive Sensing – Rice University

 Edit (15/02/15) : Corrected inaccuracies of the content.

92 thoughts on “l0-Norm, l1-Norm, l2-Norm, … , l-infinity Norm

  1. This article helps me a lot!
    There are many proper nouns that appear in papers frequently and now I finally understand the meaning of these nouns.

  2. Great article! If it just would have been that clear during my pattern recognition lectures…….

  3. Great explanation. I have also seen the use of L2/3-NORM in some Compressed Sensing work I just read and wondered if you wanted to expand on why this might be used.

  4. Could anyone please tell me how L1 norm gives sparse solutions or L1 norm is best suitable for sparse solutions?

    I also read somewhere that, more is the norm value (such as, L1, L2,L3….) more it tries to fit the outliers. What quantity in the mathematical expression of the norms makes it to behave like that?

    Thank you.

    Praveen

    1. Very informative.It has helped me a lot.Kindly mail me more information about L1 norm.

      Thanks

  5. perfect understanding that is why clear explanation is given… thank you for this nice interpretation

  6. Stumbled upon this blog when I came across matrix norm in papers I read, and, since I am now in the early phase of my PhD, I’m in a way happy if I find other PhD students in other part of the world having blogs😀. I’m even thinking to write more meaningful blog posts. Anyway, good luck!😀

  7. A complicated thing made simple… Did not understand the concept till now , but now its too clear.

  8. Very informative.It has helped me a lot.

    Kindly mail me more information about L1 norm.

    Thanks

  9. Greatly appreciated! Thank you very very much. You may have just saved me on a qualification exam for my degree🙂

  10. can u explain norms with respect to 2D matrix?? ie L^0,L^1,L^2 norm of a matrix..and to how to find out these norms…

  11. Great article, thanks a lot!!! Question: what represents axis (x and y) in graph which shows l1 and l2 solutions?

    1. Axis x and y represent 2 elements (x1,x2) of a tuple (2-dimensional vector) while the blue line is the set of possible solution of a system of equation on a plane.

  12. Please make it clear to your readers that the l0 norm *is not a norm*. It satisfies only two of the three necessary properties of a norm: that it is zero only for the all-zero vector, and the triangle inequality. It is not, however, positively homogeneous. And unlike all true norms, it is not convex. Calling it a “norm” sews confusion. It is the *cardinality function*.

  13. Awesome! So the norm’s optimisation condition is only subject to linear equation like Ax+b=0? How about linear inequality or non-linear equality?

  14. Thank you very much for the article! It gives a gentle introduction to the subject – very helpful after all those unfamiliar painful mathematical expressions I ran into.

  15. This is so very useful. Thank you very much. By the way, what is the exact application of L1-norm in optimization problems? You said applying L0-norm induce sparsity to the solution. What about L1 norms? Thank you so very much.

    1. Hi, I’m glad that you find this useful.
      The most obvious application for the L1-norm is to replace the L0-norm problem. While minimising the L0-norm is literally maximising the sparsity, the problem itself is very hard to solve using any algorithms. L1-norm problem on the other hand has many efficient solvers available.

  16. This article cleared up the L infinity norm for me, so thank you for that!
    I do want to mention that NP-hard doesn’t mean that its necessarily difficult to solve. It may be difficult to solve, may be easy to solve but difficult to solve efficiently, or not even be solvable (not decidable for example). I’m not sure the details of the particular problem you mentioned, I’m just pointing out that NP-hard is more subtle than saying its problems are complex or hard to solve.

  17. Nice explanation. one point I did not understand is the point that you mention about the matrix system being underdetermined in case of the L2 optimization. Will be a great help if you could clarify.

    Thanks.

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