Lateral excitation: Notes on computational neuroscience.
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Wrangling quartics
The problem we have is to minimize $${1\over 2} \|A ^T Z^T Z A – B \|_F^2 + {\lambda \over 2} \|Z – I\|_F^2,$$ where $B$ is symmetric. This is quartic over $Z$ so its solution will be the roots of some cubic function of $Z$, which is likely intractable. However, this problem might have…
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Nearest orthonormal matrix
We sometimes need to find the nearest orthonormal matrix $Q$ to a given square matrix $X$. This can easily be done using Lagrange multipliers, as I’ll show here. By ‘nearest’ we will mean in the element-wise sum-of-squares sense. This is equivalent to the squared Frobenius norm, so our optimization problem is $$ \argmin_Q \; {1…
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Estimating the intrinsic dimensionality of data with the participation ratio
Many datasets are samples of the values of a given set of $N$ features. We can visualise these data as points in an $N$-dimensional space, with each point corresponding to one of the samples. Visualization encourages geometric characterization. A basic geometric property is dimensionality: what is the dimension of the space in which the data…
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The nearest dataset with a given covariance
When analyzing neural activity I sometimes want to find the nearest dataset to the one I’m working with that also has a desired covariance $\Sigma$. In this post I’m going to show how to compute such a dataset. Let the dataset we’re given be the $m \times n$ matrix $X$, and the new dataset that…
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