Questions tagged [math]

For questions about mathematics related to artificial intelligence.

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39 views

Questions about a research paper on salient region detection and segmentation

I am reading this paper in an attempt to recreate the salient region detection and segmentation model employed. I have the following questions pertaining to section 3 of the paper and I would highly ...
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12 views

Is there any closed form analytical expression to represent fractional max pooling?

There are Nineteen pooling layers in PyTorch. Almost all of the layers are provided with corresponding analytical formulae. But analytical formulae is not provided for the fractional max pooling ...
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24 views

How can I compute a mathematical formula for my CNN?

Let's say, for example, I have built the following CNN model using Keras: ...
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1answer
69 views

Is it possible to know the distance objects are from camera based on only knowing one object's height?

I am doing a project where I have to know distance a particular object is from camera. In the photo I only know one of the object's height, but I don't know how far away that object is and I don't ...
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1answer
29 views

How to interpret the policy gradient expression in reinforcement learning?

I'm currently going through the OpenAI's spinning up introduction course to reinforcement learning. On one of the final sections, they derive an expression for the gradient of the undiscounted return ...
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1answer
49 views

Why is tanh a “smoothly” differentiable function?

The sigmoid, tanh, and ReLU are popular and useful activation functions in the literature. The following excerpt taken from p4 of Neural Networks and Neural Language Models says that ...
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0answers
23 views

(Deep) feature engineering for lambda terms (mathematical expressions, higher order logic formulas) - is such thing?

Automated theorem proving with (deep) reinforcement learning (DRL) approach is hot topic in current AI research when domains of games are becoming saturated and completed research topics. For example, ...
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1answer
53 views

What is the meaning or implications of the rank of a dataset for machine learning algorithms?

Consider a dataset with $n$ training examples and $d$ features. Let $D_{n \times d}$ be the data matrix and $r$ be the rank of it. In matrices, rank $r$ is generally useful in Knowing the dimension ...
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1answer
75 views

What math should I learn before and while using and applying deep learning?

I want to learn deep learning. After researching a little, I came to the conclusion that I need a lot of math. I've started a linear algebra course, and it takes a long time (2-3 weeks). I want to ...
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3answers
1k views

What are the mathematical prerequisites for an AI researcher?

What are the mathematical prerequisites for understanding the core part of various algorithms involved in artificial intelligence and developing one's own algorithms? Please, refer some specific books....
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0answers
22 views

Derivation of an probability expansion used in Word2Vec classifier model

We are using the following notations, for this question, to calculate the probability values \begin{array}{|c|c|} \hline \text{$w$} & \text{target word embedding vector} \\ \hline \text{$c$} &...
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2answers
206 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 ...
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1answer
47 views

Why don't integrated gradients explain samples correctly?

I have a linear tabular dataset made of floats. The dataset follows a simple rule like: ...
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17 views

Are there any good references that describe the equations of the forward pass of Graph Neural Networks?

I am trying to program Graph Neural Network from scratch. Can the community please suggest a good reference/s to read about the equations of the forward pass in Graph Neural Networks, especially in ...
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1answer
128 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 ...
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1answer
73 views

Is it okay to think of any dataset in artificial intelligence as a mathematical set?

A dataset is a collection of data points. It is known that the data points in the dataset can repeat. And the repetition does matter for building AI models. So, why does the word dataset contain the ...
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1answer
78 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 $\theta$?...
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2answers
89 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 ...
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1answer
31 views

how to go from mathematical problem to neural network (and back)?

I am a little confused on how, you can find online papers that describe complex Machine Learning formulas in a mathematical/probabilistic way, and, in the other hands, easy tutorials that teach you ...
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1answer
63 views

Why can a neural network use more than one activation function?

From trying to understand neural networks better, I've come upon a tentative notion that an activation function aims to build a function it's approximating via linear combinations with biases and ...
3
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1answer
159 views

How is the state-value function expressed as a product of sums?

The state-value function for a given policy $\pi$ is given by $$\begin{align} V^{\pi}(s) &=E_{\pi}\left\{r_{t+1}+\gamma r_{t+2}+\gamma^{2} r_{t+3}+\cdots \mid s_{t}=s\right\} \\ &=E_{\pi}\...
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1answer
24 views

How is the variational lower bound for hard attention derived in Show, Attend and Tell

How is the jump from line 1 to line 2 done in equation 10 of Show, Attend and Tell? While we're at it, another thing that might be muddying the waters for me is that I'm not clear on what the sum is ...
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1answer
59 views

Explanation of this L2 minimization equation

I am trying to understand the last two lines of this math notation. How Var and double summation of Cov came to the equation. The first two lines I understood something like $(a-b)^2 = a^2 -2ab +b^2$.
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1answer
51 views

Assumptions of a Linear Regression [closed]

I was going through the concept of Linear Regression and ran into the concept of deciding whether a Linear Regression Model is the best fit for your data by 5 assumptions: Linearity Homoscedasticity ...
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0answers
24 views

Do $V_\theta$ and $V_s$ represent partial or total derivatives in the paper “Learning Continuous Control Policies by Stochastic Value Gradients”?

I was reading up on the Stochastic Value Gradients paper by Heess et al. In the paper, they describe a recursive process to calculate path-wise derivatives via equations (3) and (4), at the bottom of ...
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17 views

How do we get from entropy to KL divergence in this paper?

I'm reading through Regularizing Neural Networks By Penalizing Confident Output Distributions and I'm stuck on the equation in section 3.2. It's not clear to me at all that the self-entropy of the ...
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43 views

What are the math theorems regarding the Multilayer Perceptron?

I've come across a theorem "Convergence theorem Simple Perceptron" for the first time, here-> https://zaguan.unizar.es/record/69205/files/TAZ-TFG-2018-148.pdf, page 27, (is in Spanish) ...
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1answer
223 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, ...
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2answers
54 views

Machine mathematical reasoning by clever substitutions, How to do with AI

I have three equations that relates five variables {a, b, c, r, s} with a sum and two ratios. Eq. 1: a = b + c; Eq. 2: s = b / a; Eq. 3: r = b / c. Given two values for any of the five variables I ...
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1answer
52 views

Is there a full and precise formulation of Theorem 1 in the Integrated Gradients paper?

Theorem 1 (page 5) in the paper about Integrated Gradients states that Integrated gradients is the unique path method that is symmetry-preserving. What I miss is A precise formulation of the ...
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0answers
33 views

Could the inputs of the mean squared-error loss function be transformed to allow larger learning rates?

In the context of a neural network $\hat{y} = f_\theta(\mathbf{x})$ with parameters $\theta$ that is trained to perform regression such that the prediction $\hat{\mathbf{y}} = [\hat{y}_1,\hat{y}_2,...,...
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1answer
31 views

What's the difference between a 1d tensor and a 2d tensor with 1 dimension?

I'm doing a TensorFlow tutorial, where they convert an array of the numbers [1,2,3] to a tensor like this: ...
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0answers
18 views

Clonal operator in Immune Clonal Strategy

I was reading about Immune Clonal Strategy, specifically about Monoclonal operator from Immunity clonal strategies, and it goes as follows: Here $a_i $ is a point and $a_i = \{ x_1, x_2, \cdots, x_m \...
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0answers
12 views

Is there an arrow missing in the derivation of front-door adjustment formula from do-calculus?

Here is Judea Pearl's derivation of the front-door adjustment formula: Is there an arrow from Genotype to Cancer missing in the second diagram at the right? just like this?
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29 views

Meaning of grad_outputs in torch.autograd.grad for complex input and output

Let's say we have a mathematical expression, $$ \mathbf{y} = \mathbf{Ax}, $$ where $\mathbf{y}$ and $\mathbf{x}$ are a vector, and $\mathbf{A}$ is a matrix. Let's say the vector $\mathbf{y}$ is used ...
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1answer
296 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 ...
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1answer
48 views

Is the policy gradient expression in Fundamentals of Deep Learning wrong?

I don't understand the policy gradient as explained in Chapter-9 (Deep Reinforcement Learning) of the book Fundamentals of deep learning. Here is the whole paragraph: Policy Learning via Policy ...
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1answer
62 views

Why the optimal Bellman operator of a Q-function can be approximated by a single point

I am currently studying reinforcement learning, especially DQN. In DQN, learning proceeds in such a way as to minimize the norm (least-squares, Huber, etc.) of the optimal Bellman equation and the ...
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2answers
668 views

Is there any scientific/mathematical argument that prevents deep learning from ever producing strong AI?

I read Judea Pearl's The Book of Why, in which he mentions that deep learning is just a glorified curve fitting technology, and will not be able to produce human-like intelligence. From his book ...
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2answers
698 views

What does the argmax of the expectation of the log likelihood mean?

What does the following equation mean? What does each part of the formula represent or mean? $$\theta^* = \underset {\theta}{\arg \max} \Bbb E_{x \sim p_{data}} \log {p_{model}(x|\theta) }$$
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1answer
472 views

Why is the denominator ignored in the Bayes' rule?

The naïve Bayes' generative algorithm is often represented by the following formula: $$\text{argmax}_{y} p(y|x) = \text{argmax}_y \frac{p(x|y)p(y)}{p(x)} \approx \text{argmax}_y p(x|y)p(y)$$ Why do we ...
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3answers
1k views

Can we get the inverse of the function that a neural network represents?

I was wondering if it's possible to get the inverse of a neural network. If we view a NN as a function, can we obtain its inverse? I tried to build a simple MNIST architecture, with the input of (784,...
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4answers
528 views

What is the fundamental difference between an ML model and a function?

A model can be roughly defined as any design that is able to solve an ML task. Examples of models are the neural network, decision tree, Markov network, etc. A function can be defined as a set of ...
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1answer
82 views

Can we use ML to do anything else other than predicting (in the case of mathematical problems)?

(The math problem here just serves as an example, my question is on this type of problems in general). Given two Schur polynomials, $s_\mu$, $s_\nu$, we know that we can decompose their product into a ...
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3answers
4k views

What are the mathematical prerequisites to be able to study artificial general intelligence?

What are the mathematical prerequisites to be able to study artificial general intelligence (AGI) or strong AI?
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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-...
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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 ...
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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 ...
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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 ...
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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 ...

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