Questions tagged [math]

For questions about mathematics related to artificial intelligence.

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29 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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28 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
21 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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1answer
43 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 ...
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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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17 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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11 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
36 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
39 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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1answer
47 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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1answer
73 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
66 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-...
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29 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 ...
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1answer
56 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-...
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15 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 ...
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4answers
489 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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41 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 ...
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1answer
81 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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1answer
45 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
55 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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0answers
32 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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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 ...
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0answers
10 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\}_{...
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1answer
77 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 ...
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1answer
61 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
219 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 ...
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1answer
102 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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1answer
141 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 ...
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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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1answer
48 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 ...
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1answer
70 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 ...
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0answers
42 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 ...
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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 ...
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1answer
52 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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2answers
120 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{...
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1answer
61 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 ...
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0answers
63 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
146 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 ...
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1answer
93 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)}\...
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1answer
45 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 ...
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0answers
152 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 ...
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0answers
44 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 ...
5
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1answer
152 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 ...
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0answers
73 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} \...
2
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1answer
124 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 ...
4
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1answer
92 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 ...
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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
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1answer
82 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 ...
4
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1answer
69 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. ...
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1answer
44 views

Why are all weights of a neural net updated and not just the weights of the first layer

Why are all weights of a neural net updated and not only the weights of the first hidden layer? The error-influence of the prediction by the weights of a neural net is calculated using the chain rule....