Questions tagged [math]

For questions about mathematics related to artificial intelligence.

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3
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1answer
76 views

How do you calculate KL divergence on a three-dimensional space for a Variational Autoencoder?

I'm trying to implement a variational auto-encoder (as seen in Section 3.1 here: https://arxiv.org/pdf/2004.06271.pdf). It differs from a traditional VAE because it encodes its input images to three-...
0
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1answer
55 views

Do the rows of the design matrix refer to the observations or predictors?

I attempt to understand the formulation of dictionary learning for this paper: Depression Detection via Harvesting Social Media: A Multimodal Dictionary Learning Solution Multimodal Task-Driven ...
3
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0answers
103 views

What is the meaning of the words 'bias' and 'variance' in RL?

In reinforcement learning approaches, like temporal-difference (TD) learning or Monte Carlo methods, two of the metrics used to measure their performance are the bias and the variance. What do these ...
14
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3answers
2k views

What sort of mathematical problems are there in AI that people are working on?

I recently got a 18-month postdoc position in a math department. It's a position with relative light teaching duty and a lot of freedom about what type of research that I want to do. Previously I was ...
4
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2answers
167 views

Mathematical foundations of the ability to learn

I am an undergraduate student in applied mathematics with an interest in artificial intelligence. I am currently exploring topics where I could do research. Coming from a mathematical background I am ...
1
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1answer
71 views

How to compute the gradient of the cross-entropy loss function with respect to the parameters with softmax activation function?

I've seen plenty of examples of people doing Sigmoid + MSE backpropagation implementations, yet I do not seem to understand how to implement backpropagation as stated in the title in the case of multi-...
0
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0answers
16 views

Is parameter sharing in AlBERT akin to repeated application of same function on input?

I read the AlBERT and saw that its architecture used "Parameter Sharing" among layers of the encoder. They mentioned that this was done to save model space, make fewer training parameters ...
9
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1answer
235 views

How does the forget layer of an LSTM work?

Can someone explain the mathematical intuition behind the forget layer of an LSTM? So as far as I understand it, the cell state is essentially long term memory embedding (correct me if I'm wrong), ...
2
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3answers
121 views

What does the formula $1-\sum_i(e_i-a_i)^2$ mean in this NEAT Python API?

I have looked at the documentation for the NEAT Python API found here, where it's written The error for each genome is $1-\sum_i(e_i-a_i)^2$ I have not yet learned calculus, so I can't understand ...
2
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1answer
62 views

What is $ \nabla_{\theta_{k-1}} \theta_{k}$ in the context of MAML?

I am attempting to fully understand the explicit derivation and computation of the Hessian and how it is used in MAML. I came across this blog: https://lilianweng.github.io/lil-log/2018/11/30/meta-...
4
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1answer
120 views

Why is my derivation of the back-propagation equations inconsistent with Andrew Ng's slides from Coursera?

I am using the cross-entropy cost function to calculate its derivatives using different variables $Z, W$ and $b$ at different instances. Please refer image below for calculation. As per my knowledge, ...
1
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1answer
331 views

Understanding the derivation of the first-order model-agnostic meta-learning

According to the authors of this paper, to improve the performance, they decided to drop backward pass and using a first-order approximation I found a blog which discussed how to derive the math ...
1
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0answers
33 views

Are monotonically increasing functions easier to learn?

A monotonically increasing function is a function that as x gets bigger so does its output. So, if plotted, it will never go down. Although the outputs might stay constant. Logically this seems like ...
7
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2answers
94 views

Interpretation of inverse matrix in mean calculation in Gaussian Process

The formula for mean prediction using Gaussian Process is $k(x_*, x)k(x, x)^{-1}y$, where $k$ is the covariance function. See e.g. equation 2.23 (in chapter 2) from Gaussian Processes for Machine ...
3
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1answer
114 views

Why is the equation $r(s', a, s') =\sum_{r \in \mathcal{R}} r \frac{p\left(s^{\prime}, r \mid s, a\right)}{p\left(s^{\prime} \mid s, a\right)}$true?

I am referring to eq. 3.6 (page 49) based on Sutton's online book and can be found in an image below. I could not make sense of the final derivation of the equation $r(s, a, s')$. My question is ...
2
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1answer
336 views

Which matrix represents the similarity between words when using SVD?

Two words can be similar if they co-occur "a lot" together. They can also be similar if they have similar vectors. This similarity can be captured using cosine similarity. Let $A$ be a $n \times n$ ...
5
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1answer
100 views

Why is the change in cost wrt bias in neural network equal to error in the neuron?

While reading the book on neural networks by Michael Nielson, I had a problem understanding equation (BP3), which is $$ \frac{\partial C}{\partial b_{j}^{l}}=\delta_{j}^{l} \tag{BP3}\label{BP3}, $$ ...
0
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0answers
43 views

Can I solve the below functional equation using neural networks?

I recently watched this video, in which he solves the equation $$f(x)+f\left(\frac{1}{1-x}\right) = x$$ The answer is $$f(x) = \frac{x^3-x+1}{2x(x-1)}$$ I tried to solve this functional equation using ...
2
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1answer
131 views

Does Gödel's second incompleteness theorem put a limitation on artificial intelligence systems?

According to Brian Cantwell Smith no calculation without representation Therefore, computers depend on models. So, we can say that AI is limited internally by the model and externally by the ...
27
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4answers
4k views

Can neural networks be used to prove conjectures?

Imagine I have a list (in a computer-readable form) of all problems (or statements) and proofs that math relies on. Could I train a neural network in such a way that, for example, I enter a problem ...
1
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1answer
72 views

What does $r : \mathcal{S} \times \mathcal{A} \rightarrow \mathbb{R}$ mean in the article Hindsight Experience Replay, section 2.1?

Taken from section 2.1 in the article: We consider the standard reinforcement learning formalism consisting of an agent interacting with an environment. To simplify the exposition we assume that the ...
5
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1answer
2k views

Why does the “reward to go” trick in policy gradient methods work?

In the policy gradient method, there's a trick to reduce the variance of policy gradient. We use causality, and remove part of the sum over rewards so that only actions happened after the reward are ...
2
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2answers
124 views

How does PCA work when we reduce the original space to 2 or higher-dimensional space?

How does PCA work when we reduce the original space to a 2 or higher-dimensional space? I understand the case when we reduce the dimensionality to $1$, but not this case. $$\begin{array}{ll} \text{...
5
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1answer
274 views

Which functions can be activation functions?

What are the required characteristics of an activation function (in a neural network)? Which functions can be activation functions? For example, which of the functions below can be used as an ...
6
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1answer
391 views

What is the mathematical definition of an activation function? [duplicate]

What is the mathematical definition of an activation function to be used in a neural network? So far I did not find a precise one, summarizing which criterions (e.g. monotonicity, differentiability, ...
3
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1answer
226 views

Why does the variational auto-encoder use the reconstruction loss?

VAE is trained to reduce the following two losses. KL divergence between inferred latent distribution and Gaussian. the reconstruction loss I understand that the first one regularizes VAE to get ...
1
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1answer
118 views

How to mathematically describe the convolution operation (with a Gaussian kernel)?

I have to build a model where I pre-process the data with a Gaussian kernel. The data are an $n\times n$ matrix (i.e one channel), but not an image, thus I can't refer to this matrix as an image and ...
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0answers
35 views

Can any area of math come into play in Machine Learning Research?

As I read online following areas in mathematics comes into play in ML research Linear Algebra Calculus Differential Equations Probability Statistics Discrete Mathematics Optimization Analytic ...
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3answers
2k views

What are examples of applications of the Fourier transform to AI?

The (discrete and continuous) Fourier transform (FT) is used in signal processing in order to convert a signal (or function) in a certain domain (e.g. the time domain) to another domain (e.g., ...
1
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1answer
110 views

What is the definition of the “cost” function in the SVM's objective function?

In a course that I am attending, the cost function of a support vector machine is given by $$J(\theta)=\sum_{i=1}^{m} y^{(i)} \operatorname{cost}_{1}\left(\theta^{T} x^{(i)}\right)+\left(1-y^{(i)}\...
1
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0answers
32 views

How do I derive the gradient of the log-likelihood of an RBM?

In a Restricted Boltzmann Machine (RBM), the likelihood function is: $$p(\mathbf{v};\mathbf{\theta}) = \frac{1}{Z} \sum_{\mathbf{h}} e^{-E(\mathbf{v},\mathbf{h};\mathbf{\theta})}$$ Where $E$ is the ...
1
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1answer
111 views

The mathematics in the CBOW and Skip-Gram models

this is my first question on AI Stack Exchange. I am a mathematics student who is learning NLP so I have paid a high amount of attention on the mathematics used in the subject, but my interpretations ...
1
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0answers
11 views

How do I find the data-point with respect to a given frame?

I've been reading this paper that formulates invariant task-parametrized HSMMs. In section 3.1 (Model Learning), the task parameters are represented in $F$ coordinate systems defined by $\{A_j,b_j\}_{...
2
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1answer
297 views

Why would the lookup table (of a table-driven artificial agent) need to store data at pixel precision?

While reading the book AI A modern approach, 4th ed, I came across the section of "Agent program" with following text: It is instructive to consider why the table-driven approach to agent ...
6
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1answer
153 views

Formal definition of the Object Detection problem

For many problems in computer science, there is a formal, mathematical problem defition. Something like: Given ..., the problem is to ... How can the Object Detection problem (i.e. detecting objects ...
3
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0answers
21 views

Mapping given probabilities to empirical probabilities

Consider following problem statement: You have given $n$ actions. You can perform any of them. Each action gives you success with some probability. The challenge is to perform given finite number of ...
2
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0answers
164 views

Choice of fuzzification function

I'm a relative newbie to fuzzie logic systems but I have some knowledge in mathematics. I have the following problem: I want to fuzzify certain values. Some are in the range [-$\inf$,$\inf$] and some ...
2
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1answer
53 views

Research paths/areas for improving the performance of CNNs when faced with limited data

I've been reading through the research literature for image processing, computer vision, and convolutional neural networks. For image classification and object recognition, I know that convolutional ...
1
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0answers
45 views

Could the neural network automatically calculate and get different one-to-many quantities relative to their parent quantity?

Let's say I have a primary dataset that its secondary dataset is hundreds to match and group like an one-to-many relationship. I'm new in this world of the AI but my problem is that many child groups ...
0
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0answers
30 views

Confusion on Math Notation Definition

I attempt to understand the formulation of dictionary learning for this paper: Depression Detection via Harvesting Social Media: A Multimodal Dictionary Learning Solution Multimodal Task-Driven ...
13
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13answers
2k views

How should I get started with artificial intelligence? [duplicate]

What is the mathematical background required to start learning AI? What else should I also learn?
0
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1answer
73 views

Mathematical calculation behind decision tree classifier with continuous variables

Problem Description I am working on a binary classification problem having continuous variables (Gene expression Values). My goal is to classify the samples as case ...
20
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1answer
7k views

What is the Bellman operator in reinforcement learning?

In mathematics, the word operator can refer to several distinct but related concepts. An operator can be defined as a function between two vector spaces, it can be defined as a function where the ...
0
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1answer
120 views

What is Bayes' theorem?

What is Bayes' theorem? How does it relate to conditional probabilities?
2
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1answer
170 views

Understand the DDPG algorithm in Keras

I'm trying to understand the DDPG algorithm using Keras I found the site and started analyzing the code, I can't understand 2 things. The algorithm used to write the code presented on the page In the ...
2
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4answers
378 views

Can we define the AI singularity mathematically?

The "AI Singularity" or "Technological Singularity" is a vague term that roughly seems to refer to the idea of: Humans can design algorithms Humans can improve algorithms Eventually algorithms we ...
1
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1answer
48 views

How to understand mapping function of kernel?

For a kernel function, we have two conditions one is that it should be symmetric which is easy to understand intuitively because dot products are symmetric as well and our kernel should also follow ...
2
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3answers
237 views

Understanding a few terms in Andrew Ng's definition of the cost function for linear regression

I have completed week 1 of Andrew Ng's course. I understand that the cost function for linear regression is defined as $J (\theta_0, \theta_1) = 1/2m*\sum (h(x)-y)^2$ and the $h$ is defined as $h(x) = ...
2
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1answer
147 views

Can you help me understand how weight normalization works?

I am trying to dissect the paper Weight Normalization: A Simple Reparameterization to Accelerate Training of Deep Neural Networks. Unfortunately, because my math is a little bit rusty, I got a little ...
4
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2answers
872 views

How good is AI in math?

Currently, AI is advancing fast in deep learning: Entire human chess knowledge learned and surpassed by DeepMind's AlphaZero in four hours. As a layman, I'm taking this as a quite powerful searching ...