# Questions tagged [machine-learning]

How can we build computer systems that automatically improve with experience, and what are the fundamental laws that govern all learning processes?

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### Why the left matrix of single value decomposition obtained by scipy.linalg.svd() and eigenvectors obtained from scipy.linalg.eig() are different?

I tried to get eigen vectors by single value decomposition using scipy.linalg.svd() which gives me three outputs the left matrix ...
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### Mathematics for Machine Learning by Marc Peter Deisenroth, A. Aldo Faisal, and Cheng Soon Ong. Theorem 2.20 (basis change) questions

I am struggling to understand this proof, and have some questions which I will list. Here is a link to the textbook https://mml-book.github.io/book/mml-book.pdf and the theorem and proof I am ...
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### Why is the base of the exponential function in the softmax formula chosen as e, rather than 2 or other numbers?

I have been studying the softmax function, and I am curious why the natural constant $e$ is used as the base of the exponential function in the softmax formula. Specifically, I would like to ...
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### Barto and Sutton's Reinforcement Learning Exercise 3.11: Don't understand solution because of knowledge gaps

I'm going through Barto and Sutton's Reinforcement Learning book with only knowledge of high-school Pre-calc and previous programming experience. I've been able to go through Chapters 1 and 2 ...
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### Lipschitz Continuous Function in the Embedding Space

Given three metric spaces $(X,d_X),(Y,d_Y),(Z,d_Z)$ and an embedding function $g: X \rightarrow Z$, I am interested in finding more information about functions $f: X \rightarrow Y$ in which there ...
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### Approximating inverse of an ill-conditioned matrix

I am not sure whether this is the right place to ask this question or not, it might be not really a mathematical question but more a deep learning question. But the base of question is mathematics, so ...
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### How to estimate the inverse of a non-invertible matrix?

So I'm working on a machine learning problem where my solution requires taking the inverse of a matrix at some point. The problem is that this matrix is sometimes non-invertible. In theory the the ...
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### Formulating a solution ansatz for the 1D heat equation in polar coordinates to learn the PDE in a PINN setting

Hello Math Stack Exchange Community, I am working on solving a partial differential equation (PDE) with a neural network in a PINN-like fashion, and I am seeking advice on identifying an appropriate ...
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### What is the collection of functions that a given finite neural network can approximate with ease?

To my understanding, one version of the universal approximation theorem runs as follows: Let $\Phi$ be the family of (trained) feedforward neural networks of bounded width, arbitrary depth, and mild ...
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### An upper bound of the sum of factorial in Vapnik and Chervonenkis, 1971

How to prove the following inequality: \begin{align*} \Gamma=\sum_{k\in[0,m]\wedge|\frac{2k}{l}-\frac{m}{l}|\ge\frac{\epsilon}{2}}\frac{{m\choose k}{2l-m\choose l-k}}{{2l\choose l}}\le2e^{-\frac{\...
Im a little confused with the likelihood function. For discriminative models, we have a hypothesis function $h_{\theta}(x) = p(y \mid x ; \theta)$. Using the principles of maximim likelihood we want ...