Throughout this post, will explore some basic use of Singular Value Decompositionone of the most common and powerful linear algebra concepts used in all sub-fields of computer science, from said image processing, through recommendation systems, ML, to ranking web pages in search engines.
We will see how it can be used to analyze 2D image processing filters — check if they are separable, approximate the non-separable ones, and will demonstrate how to use those for separable, juicy bokeh. Note: this post turned out to be a part one of a mini-series, and has a follow-up post here — be sure to check it out! For large filters, this can get easily prohibitively expensive, and we get quadratic scaling with the filter spatial extent… This is where separable filters can come to the rescue.
If a filter is separable, we can decompose such filter into a sequence of two 1D filters in different directions usually horizontal, and then vertical. This requires storing the intermediate results — either in memory, or locally line buffering, tiled local memory optimizations. While you pay the cost of storing the intermediate results and synchronizing the passes, you get linear and not quadratic scaling.
Two simplest filters that are bread and butter of any image processing are the box filterand the Gaussian filter. Both of them are separable, but why? This last product is the equation of separability — our 2D function is a product of a horizontal, and a vertical function. We started with two trivial, analytical filters, but what do we do if we have an arbitrary filter given to us in a numerical form — just a matrix full of numbers?
We can plot them, analyze them, but how do we check if given filter is separable? Can we try to separate maybe at least approximately any filter? Given a matrix MxN, we would like to express it as a product of two matrices, Mx1 and 1xN. There are a few ways of looking at this problem they are roughly equivalent — but writing them all out should help build some intuition for it :. To solve this problem, we will use one of the most common and useful linear algebra tools, Singular Value Decomposition.
How does SVD help us here? If our original matrix is separable, then it is rank 1, and we will have only single singular value. Even if they are not zero but significantly smallerand we truncate all coefficients except for the first one, we will get a separable approximation of the original filter matrix! Computing SVD efficiently and numerically stable is generally difficult requires lots of careful consideration and I am not going to cover it — but almost every linear algebra package or library has it implemented, and I assume we just use one.
This is just a warm-up, so feel free to skip those two trivial cases and jump to the next section.Siemens hmi
Great, we get only one singular value — which means that it is a rank 1 matrix and fully separable. We can also get rid of the singular value and embed it into our filters, for example bymultiplying both by sqrt of it to make them comparable in value:.This article uses the SVD to construct a low-rank approximation to an image. Applications include image compression and denoising an image.
The value of each pixel in a grayscale image can be stored in a matrix where each element of the matrix is a value between 0 off and 1 full intensity.Computing the Singular Value Decomposition - MIT 18.06SC Linear Algebra, Fall 2011
I want to create a small example that is easy to view, so I'll create a small matrix that contains information for a low-resolution image of the capital letters "SVD. Because the data matrix contains only five non-zero rows, the rank of the A matrix cannot be more than 5. The following statements compute the SVD of the data matrix and create a plot of the singular values. The plot of the singular values is similar to the scree plot in principal component analysisand you can use the plot to help choose the number of components to retain when approximating the data.
For this example, it looks like retaining three or five components would be good choices for approximating the data. To see how low-rank approximations work, let's generate and view the rank-1, rank-2, and rank-3 approximations:. The rank-1 approximation does a good job of determining the columns and do not contain that contain the letters.
The approximation also picks out two rows that contain the horizontal strokes of the capital "S. The rank-2 approximation refines the image and adds additional details.
You can begin to make out the letters "SVD. The rank-3 approximation contains enough details that someone unfamiliar with the message can read it. The "S" is reconstructed almost perfectly and the space inside the "V" and "D" is almost empty.
Even though this data is five-dimensional, this three-dimensional approximation is very good. Not only is a low-rank approximation easier to work with than the original five-dimensional data, but a low-rank approximation represents a compression of the data. For the rank-3 approximation, three columns of the U matrix contain 33 numbers and three columns of V T contain 15 numbers.
So the total number of values required to represent the rank-3 approximation is only 48, which is almost half the number of values as for the original image. You can use the singular value decomposition and low-rank approximations to try to eliminate random noise that has corrupted an image.
Every TV detective series has shown an episode in which the police obtain a blurry image of a suspect's face or license plate.
The detective asks the computer technician if she can enhance the image. With the push of a button, the blurry image is replaced by a crystal clear image that enables the police to identify and apprehend the criminal. The image reconstruction algorithms used in modern law enforcement are more sophisticated than the SVD.
Nevertheless, the SVD can do a reasonable job of removing small random noise from an image, thereby making certain features easier to see. The SVD has to have enough data to work with, so the following statements duplicate the "SVD" image four times before adding random Gaussian noise to the data.
The noisy data is displayed as a heat map on the range [ I think most police officers would be able to read this message in spite of the added noise, but let's see if the SVD can clean it up. The following statements compute the SVD and create a plot of the singular values:. There are 14 non-zero singular values.
In theory, the main signal is contained in the components that correspond to the largest singular values whereas the noise is captured by the components for the smallest singular values.
For these data, the plot of the singular values suggests that three or five or nine components might capture the main signal while ignoring the noise. The following statements create and display the rank-3 and rank-5 approximations. Only the rank-5 approximation is shown.
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The denoised low-rank image is not as clear as Hollywood depicts, but it is quite readable. It is certainly good enough to identify a potential suspect.
I can hear the detective mutter, "Book 'em, Danno! Murder One. In summary, the singular value decomposition SVD enables you to approximate a data matrix by using a low-rank approximation.My original motivation was to play a bit with JAX and see if it will be useful for me, and I immediately had this very simple use-case in mind. This post comes with a colab that you can run and modify yourself.
In the last blog post, we have looked at analyzing if convolutional image filters can be made separable, and if not, finding separable approximations as a sum of N separable filters.
For this we used Singular Value Decomposition and using a low rank approximation by taking first N singular values. I have suggested that this can be solved with optimization, and in this post I will describe most likely the simplest method to do so! Side note: After my blog post, I had a chat with a colleague of mine, Soufiane Khiat — we used to work together at Ubisoft Montreal — and given his mathematical background much better than mine, he had some cool insights on this problem.
I still recommend reading about this approach if you want to go much deeper into the topic and again — thanks Soufiane for a great discussion. But this is not the kind of optimization that a mathematician, or most academics think of and not the topic of my blog post! Optimization that I will use is the one in which you have some function that depends on a set of parameters, and your goal is to find the set of parameters that achieves a certain goal, usually minimum sometimes maximum, but they can be equivalent in most cases over this function.
This definition is very general — and in theory it even covers also computational performance optimizations we are looking for a set of computer program instructions that optimizes performance while not diverging from the desired output. Optimization of arbitrary functions is generally a NP-hard problem there are no solutions other than exploring every possible value, which is impossible in the case of continuous functionsbut under some constraints like convexityor looking for local minimait can be made feasible, and relatively robust and fast — and is basis of modern convolutional neural networks, and algorithms like gradient descent.
This post will skip explaining the gradient descent. JAX got very popular recently might be a sampling bias, but seemed to me like half of ML twitter was talking about itand I am usually very skeptical about such new hype bandwagons. But I was also not very happy with my previous options, so I tried it, and I think it is popular for a few good reasons:.
This might seem obvious, but before we can start optimizing an objective, we have to define it in some way that is understandable for the computer and optimizeable. What is non-obvious is that coming up with a decent objective function is the biggest challenge of machine learningIMO a much bigger problem than any particular choice of a technique.
Note: it also has much wider implications than our toy problems; ill-defined objectives can lead to catastrophes in wider social context, e. We have a hard limit on maximum of N separable passes and fixed number of coefficients. But on the other hand, the two other goals are not well defined. Mathematically, we use term of distance, or metric.
Convenient and well researched are metrics that are based on p-norm and mathematicians like to operate on squared Euclidean distance L2 normso average squared errors. Average squared error is so often used as it usually has a simple, closed form solutions like linear least squareslinear regression, PCAbut in many cases it might not be the right loss. Defining perceptual similarity is the most difficult and is an open problem in computer vision, with the recent universal approach of using similarity features extracted by neural networks for the purpose of image recognition.
Deciding on components of the loss function to avoid visual artifacts and tuning them is often the most time consuming part of optimization and machine learning.
This is a very common approach — summing together multiple different terms with different meanings and finding the parameters that optimize such a sum. Open challenge is tuning the weights of the different components of the loss function, a further section will show the impact of it.
This loss function might not be the best one, but this is what makes such problems fun — often designing a better loss closer corresponding to our intentions can lead to significantly better results without changing the algorithm at all!
This was one of my huge surprises — so many great papers just propose simple optimization framework together with an improved loss function to advance state of the art significantly. Now that we have a loss function to optimize, we need to find parameters that minimize it. This is where things can become difficult. If we wanted to do a brute force search in this dimensional space, this could take a very long time!
Luckily, we are in a situation where our initial guess obtained through SVD is already kind of okand we want to just improve it. How do we compute the gradient of our loss function? This is where various auto-differentiation libraries can help us. This can be achieved either symbolically, or in some cases even numerically if closed-form gradient would be impossible to compute.
JAX can be a drop-in replacement to a combo of pure Python and numpy, keeping most of the functions exactly the same! In colab, you can import it either instead of numpy, or in addition to numpy.
It only takes a minute to sign up. Let me try. No, generally those two matrices are different. Here is my implementation of that method in Python using the Numpy library for the typical case where the columns of A are linearly independent:. Sign up to join this community.
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The Overflow Blog. Socializing with co-workers while social distancing. Featured on Meta. Community and Moderator guidelines for escalating issues via new response….In mathematics, low-rank approximation is a minimization problem, in which the cost function measures the fit between a given matrix the data and an approximating matrix the optimization variablesubject to a constraint that the approximating matrix has reduced rank.
The problem is used for mathematical modeling and data compression. The rank constraint is related to a constraint on the complexity of a model that fits the data. In applications, often there are other constraints on the approximating matrix apart from the rank constraint, e. The unstructured problem with fit measured by the Frobenius normi.
The result is referred to as the matrix approximation lemma or Eckart—Young—Mirsky theorem. Suppose that. Prior knowledge about distribution of the errors can be taken into account by considering the weighted low-rank approximation problem.
Matrix Factorization for Movie Recommendations in Python
The general weighted low-rank approximation problem does not admit an analytic solution in terms of the singular value decomposition and is solved by local optimization methods, which provide no guarantee that a globally optimal solution is found. Such distances matrices are commonly computed in software packages and have applications to learning image manifolds, handwriting recognition, and multi-dimensional unfolding.
In an attempt to reduce their description size,   one can study low rank approximation of such matrices. The low-rank approximation problems in the distributed and streaming setting has been consider in . The resulting optimization algorithm called alternating projections is globally convergent with a linear convergence rate to a locally optimal solution of the weighted low-rank approximation problem.
The iteration is stopped when a user defined convergence condition is satisfied. Matlab implementation of the alternating projections algorithm for weighted low-rank approximation:. The bilinear nature of the problem is effectively used in an alternative approach, called variable projections. Consider again the weighted low rank approximation problem, parameterized in the image form.
For this purpose standard optimization methods, e. Matlab implementation of the variable projections algorithm for weighted low-rank approximation:.
The variable projections approach can be applied also to low rank approximation problems parameterized in the kernel form.
The method is effective when the number of eliminated variables is much larger than the number of optimization variables left at the stage of the nonlinear least squares minimization. Such problems occur in system identification, parameterized in the kernel form, where the eliminated variables are the approximating trajectory and the remaining variables are the model parameters.
In the context of linear time-invariant systemsthe elimination step is equivalent to Kalman smoothing. Usually, we want our new solution not only to be of low rank, but also satisfy other convex constraints due to application requirements. Our interested problem would be as follows.
This problem can find tons of real applications, including to recover a good solution from an inexact semidefinite programming relaxation. This problem is helpful in solving many problems. However, it is challenging due to the combination of the convex and nonconvex low-rank constraints. However, the Alternating Direction Method of Multipliers ADMM can be applied to solve the nonconvex problem with convex objective function, rank constraints and other convex constraints,  and is thus suitable to solve our above problem.
Moreover, unlike the general nonconvex problems, ADMM will guarantee to converge a feasible solution as long as its dual variable converges in the iterations. From Wikipedia, the free encyclopedia. Low-rank approximation is closely related to: principal component analysisfactor analysistotal least squareslatent semantic analysisand orthogonal regression.
Markovsky, Structured low-rank approximation and its applications, Automatica, Volume 44, Issue 4, AprilPages — Markovsky, J. Willems, S. Van HuffelB. De Moor, and R. Pintelon, Application of structured total least squares for system identification and model reduction.Facepunch model rip
Eckart, G. Young, The approximation of one matrix by another of lower rank.PreviouslyI used item-based collaborative filtering to make music recommendations from raw artist listen-count data. I had a decent amount of data, and ended up making some pretty good recommendations. Unfortunately, there are two issues with taking this approach:. I talked about the scaling issue in the previous post, but not the conceptual issue. The key concern is that ratings matrices may be overfit and noisy representations of user tastes and preferences.
Mathematically, the dot product of our action vectors would be 0. Using item features such as genre could help fix this issue, but not entirely. Stealing an example from Joseph Konston professor at Minnesota who has a Coursera course on recommender systemswhat if we both like songs with great storytelling, regardless of the genre?
How do we resolve this? I need a method that can derive tastes and preference vectors from the raw data. Matrix factorization is the breaking down of one matrix into a product of multiple matrices. There are many different ways to factor matrices, but singular value decomposition is particularly useful for making recommendations. So what is singular value decomposition SVD? At a high level, SVD is an algorithm that decomposes a matrix into the best lower rank i. Mathematically, it decomposes into two unitary matrices and a diagonal matrix:.
To get the lower rank approximation, we take these matrices and keep only the top features, which we think of as the most important underlying taste and preference vectors. These look good, but I want the format of my ratings matrix to be one row per user and one column per movie. The last thing I need to do is de-mean the data normalize by each users mean and convert it from a dataframe to a numpy array. All set. Scipy and Numpy both have functions to do the singular value decomposition.When a is a 2D array, it is factorized as u np.
When a is higher-dimensional, SVD is applied in stacked mode as explained below. A real or complex array with a. If True defaultu and vh have the shapes Otherwise, the shapes are Whether or not to compute u and vh in addition to s.
True by default. Unitary array s. The first a. Vector s with the singular values, within each vector sorted in descending order. If True, a is assumed to be Hermitian symmetric if real-valuedenabling a more efficient method for finding singular values. Defaults to False. Changed in version 1. SVD is usually described for the factorization of a 2D matrix. The higher-dimensional case will be discussed below. In the 2D case, SVD is written aswhere, and. The 1D array s contains the singular values of a and u and vh are unitary.
The rows of vh are the eigenvectors of and the columns of u are the eigenvectors of. If a has more than two dimensions, then broadcasting rules apply, as explained in Linear algebra on several matrices at once. The operator can be replaced by the function np. If a is a matrix object as opposed to an ndarraythen so are all the return values.
New in version 1. Previous topic numpy. Last updated on Jul 26, Created using Sphinx 1.
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