Linear decorrelation models
Neural Computation 2024-2025.
9th April 2025
Practical info
Organize your answers according to the questions; don’t merge them. Plots should include axis labels and units (either on the plot, or mentioned in the text), see my web page link.
There will be a to-be-determined normalization factor between the number of points scored and the resulting percentage mark.
You will find that some questions are quite open-ended. In order to receive full marks for those questions your answers need to go beyond running a simulation and making a plot. Instead, you should substantiate your explanations and claims, for instance by doing additional simulations or mathematical analysis. It should not be necessary to consult scientific literature, but if you do use additional literature, cite it.
Copying results is absolutely not allowed and can lead to severe punishment. It’s OK to ask for help from your friends. However, this help must not extend to copying code, results, or written text that your friend has written, or that you and your friend have written together. I assess you on the basis of what you are able to do by yourself. It’s OK to help a friend. However, this help must not extend to providing your friend with code or written text. If you are found to have done so, a penalty will be assessed against you as well.
Deadline will be announced via email and the website. Upload report and code used for question 3 on Moodle. Use any computer language you like.
Model
One common theory about neural processing in the retina is that it reduces correlations in the in-put. We examine linear transformations that whiten / de-correlate the inputs. Specifically, given N dimensional input vectors x and N dimensional output we look for N × N matrices W , so that the output y = Wx, is uncorrelated. We also impose that the output covariance is normalized so that it has covariance matrix ⟨yyT〉= IN , where IN is the N-dimensional identity matrix.
Question 1 (5 points) On the moodle page you will find some black and white photos of natural scenes (all_images.zip). Sample some 10000 images patches of some 10x10 pixels. Before sampling, normalize each image such that its mean pixel value is zero and pixel variance is one. Plot the covariance matrix and comment on its shape. [In Matlab/Octave you can use imread() to read image files.]
Question 2 (5 points) Use PCA on the data to find a whitening transformation. Show that the covariance matrix of the output indeed equals the identity matrix. Also plot a few (~10) of the receptive fields, and comment on their shape.
Question 3 (5 points) Another, perhaps more natural, way to whiten the input is given in the lecture notes (page 74). Implement this and show again that the covariance matrix of the output equals the identity matrix. Compare the L1 norms of the weight matrix (Σi,j|wij|) with the PCA solution. Why is the L1 norm relevant biologically? What other properties of the weight matrix could be important for biology?
Question 4 (5 points, harder) Are there whitening matrices with even lower L1 norm?
Question 5 (5 points, harder) Assume a simplified model with just 2 inputs and an input covariance What are all the matrices that de-correlate this input? Which matrices have minimum L1 norm?