Category Archives: Uncategorized

Least square fitting of vector-valued random variables

Let be joint random variables (typically not independent). This post describes which linear map relates these variables best in the sense that the expected square error is minimised. Related applications such as principle component analysis and auto encoding under random … Continue reading

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Linear autoencoder with dropout

We begin with a brief review of least squares fitting formulated in autoencoder language. Let be a random variable in such that and let be its covariance matrix. Then is a self-adjoint (symmetric) operator. Let be a positive number not … Continue reading

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Mutual information and fast image registration

In the previous post the mutual information for a pair of images was expressed in terms of the angle between the gradients of and : . Note that if both images are equal then everywhere and mutual information is not … Continue reading

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Mutual information and gradients

In this post we will see a that mutual information between functions (e.g. two images) can be expressed in terms of their gradient fields. First some definitions and background. Let be a differential function for some . In this post  typically … Continue reading

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Projections of the Mahalanobis norm

In this post is the standard inner product on for some and is a fixed Hermitian positive definite operator. The square Mahalanobis norm of a vector for this operator is defined by . The results in this post were found … Continue reading

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The quantum harmonic oscillator in the plane

This post investigates some solutions of the (or rather a) quantum harmonic oscillator in the plane. Here the plane is which will be identified with through . Let and denote the partial derivative operators with respect to and respectively. Then … Continue reading

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Bounds for central binomial coefficients

Let be a positive integer. The binomial coefficient is called a central binomial coefficient. Several bounds for these coefficients are known and the more advanced ones are commonly derived from Stirling’s formula for factorials or from Wallis’ product for . This … Continue reading

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