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Questions tagged [math]

For questions about mathematics related to artificial intelligence.

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2
votes
1answer
49 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 ...
0
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1answer
47 views

How to mathematically describe convolving operation of Gaussian kernel

I have build a model where I pre-process data with Gaussian kernel. The data are $n\times n$ matrix of one channel, but not an image, thus I cant reference to this matrix as image and to its elements ...
5
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2answers
92 views
+50

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 ...
2
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0answers
13 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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1answer
144 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 ...
1
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1answer
35 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 ...
0
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1answer
53 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 ...
0
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0answers
37 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 ...
1
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1answer
50 views

Given an axis-angle rotation vector, how can I find the unit rotation axis and angle?

I have a robotics assignment, which I am unable to solve. Given the axis-angle rotation vector $\Theta = (2, 2, 0)$, how can I calculate the unit vector of the rotation axis $k$ and the angle $\theta$?...
0
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1answer
36 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 ...
0
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0answers
23 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?
25
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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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2answers
92 views

The math behind PCA

I am trying to understand the math behind PCA. I can only solve it in the case of mapping vectors to 1 Dimensional space. How to solve the math in the case we reduce the number of dimension is greater ...
0
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1answer
33 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 ...
4
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1answer
205 views

Defining formula for fuzzy equation

I'm learning fuzzy logic and more or less understand the basic concept, but i'm having a hard time understanding how to apply it to a method. I tried browsing online for explanation on how to use it, ...
2
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1answer
77 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, ...
2
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1answer
95 views

In a single neuron output layer should the output be a scalar?

Given a neural network with 3 inputs, 4 hidden layers, and 1 output, should the output neuron be a vector or a scalar? I thought that at the end of the summation only one number between 0 and 1 would ...
13
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1answer
4k 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
114 views

What is Bayes' theorem?

What is Bayes' theorem? How does it relate to conditional probabilities?
0
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0answers
51 views

Can a computer make a proof by induction?

Can a computer solve the following problem, i.e. make a proof by induction? And why? Prove by induction that $$\sum_{k=1}^nk^3=\left(\frac{n(n+1)}{2}\right)^2, \, \, \, \forall n\in\mathbb N .$$ I'm ...
2
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1answer
66 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 ...
3
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4answers
332 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
37 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 ...
3
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3answers
223 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) = ...
6
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2answers
702 views

Why is the log probability replaced with the importance sampling in the loss function?

In the Trust-Region Policy Optimisation (TRPO) algorithm (and subsequently in PPO also), I do not understand the motivation behind replacing the log probability term from standard policy gradients $$...
1
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1answer
56 views

Formal proof that every purely reactive agent has behaviorally equivalent standard agent

It kind of makes sense intuitively but I'm not sure about a formal proof. I'll start with briefly listing definitions from Intro to Multiagent systems, Wooldridge, 2002 and then give you my reasoning ...
2
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1answer
122 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 ...
1
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1answer
70 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)}\...
4
votes
2answers
837 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 ...
2
votes
1answer
90 views

How can I learn tensors for deep learning?

I've seen in most deep learning papers use tensors. I understood what tensors are, but I want to dive into them, because I think that might be beneficial for further studies in Artificial Intelligence....
5
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1answer
153 views

How could an AI be used to improve the teaching and learning of mathematics?

I have been working with AI methods. I am thinking about how my daughter (and also other kids) could learn mathematics with the help of AI. For example, how could an AI be used to show the mistakes ...
1
vote
1answer
76 views

Why do we use the word “kernel” in the expression “Gaussian kernel”?

I've heard the expression "Gaussian kernel" in several contexts (e.g. in the kernel trick used in SVM). A Gaussian kernel usually refers to a Gaussian function (that is, a function similar to the ...
1
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2answers
207 views

In the context of importance sampling ratio, how is the equation $\mathbb{E}\left[\rho_{t: T-1} G_{t} | S_{t}=s\right]=v_{\pi}(s)$ derived?

When reading the book by Sutton and Barto, I came across the importance sampling ratio. The first equation, I believe, describes the probability a particular sequence is obtained given the current ...
15
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3answers
7k views

How to choose an activation function?

I choose the activation function for the output layer depending on the output that I need and the properties of the activation function that I know. For example, I choose the sigmoid function when I'm ...
5
votes
1answer
135 views

Can deep learning be used to help mathematical research?

I am currently learning about deep learning and artificial intelligence and exploring his possibilities, and, as a mathematician at heart, I am inquisitive about how it can be used to solve problems ...
2
votes
2answers
182 views

Why is MSE used over other quadratic loss functions?

So I was wondering, why I have only encountered square loss function also known as MSE. The only nice property of MSE I am so far aware of is its convex nature. But then all equations of the form $x^{...
4
votes
1answer
62 views

How can a single sample represent the expectation in gradient temporal difference learning?

I was reading the gradient temporal difference learning version 2(GTD2) from rich Sutton's book page-246. At some point, he expressed the whole expectation using a single sample from the environment. ...
1
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2answers
3k views

What is the derivative of the Leaky ReLU activation function?

I am implementing a feed-forward neural network with leaky ReLU activation functions and back-propagation from scratch. Now, I need to compute the partial derivatives, but I don't know what the ...
1
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0answers
63 views

What is an auto-associator?

What is an auto-associator, and how does it work? How can we design an auto-associator for a given pattern? I couldn't find a clear explanation for this anywhere on the internet. Here's an example of ...
1
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0answers
43 views

Simplifying Log Loss

I am reading through a paper (https://www.mitpressjournals.org/doi/pdf/10.1162/0891201053630273) where they describe logloss as a ranking function and can be simplified to the margin of the training ...
2
votes
2answers
99 views

Is ReLU a non-linear activation function?

According to this blog post The purpose of an activation function is to add some kind of non-linear property to the function The sigmoid is typically used as an activation function of a unit of a ...
2
votes
0answers
69 views

Is this the correct gradient for log of softmax? [duplicate]

I am currently implementing the very basic version (REINFORCE) of the Monte Carlo policy gradient algorithm. I was wondering if this is the correct gradient for the log of softmax. \begin{align} \...
1
vote
1answer
58 views

Why does the discriminator minimize the cross-entropy while the generator maximize it?

In his original GAN paper Goodfellow gives a game theoretic perspective for GANs: \begin{equation} \underset{G}{\min}\, \underset{D}{\max}\, V\left(D,G \right) = \mathbb{E}_{x\sim\mathit{p}_{\...
4
votes
1answer
70 views

How is the Jacobian a generalisation of the gradient?

I came across these slides Natural Language Processing with Deep Learning CS224N/Ling284, in the context of natural language processing, which talk about the Jacobian as a generalization of the ...
3
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1answer
187 views

How does one even begin to mathematically model an AI algorithm?

How does one even begin to mathematically model an AI algorithm, like alpha-beta pruning or even its thousands of variations, to determine which variation is best?
1
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0answers
54 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 external by the ...
10
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2answers
938 views

How do we prove the n-step return error reduction property?

In section 7.1 (about the n-step bootstrapping) of the book Reinforcement Learning: An Introduction (2nd edition), by Andrew Barto and Richard S. Sutton, the authors write about what they call the "n-...
1
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0answers
35 views

How do I approach this problem?

Let's say I have a dataset with multiple types of multiple ingredients (salt1,salt2, etc). Each n-th variation of each ingredient vs flavor may be represented by an n×k matrix that where an ingredient ...
2
votes
1answer
58 views

How is the log-derivative trick of a trajectory derived?

I am looking at this formula which breaks down the gradient of $P(\tau |\theta)$ the first part is clear as is the derivative of $\log(x)$, but I do not see how the first formula is rearranged into ...