Questions tagged [math]

For questions about mathematics related to artificial intelligence.

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39 views

What is Lipschitz constraint and why it is enforced on discriminator?

The following is the abstract for the research paper titled Improved Training of Wasserstein GANs Generative Adversarial Networks (GANs) are powerful generative models, but suffer from training ...
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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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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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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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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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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 ...
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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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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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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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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\}_{...
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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 ...
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182 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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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 ...
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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 ...
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0answers
21 views

How do I decide which norm to use for placing a constraint on my adversarial perturbation?

I am performing an adversarial machine learning attack on a neural network for network traffic classification. For adding adversarial perturbations in features such as packet interarrival times and ...
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1answer
300 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
43 views

What is the equation of the separation line for this neuron with identity activation?

I have a single neuron with 2 inputs, and identity activation, where f is activation function and u is output: $u = f(w_1x_1 + ...
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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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101 views

Is Gradient Descent algorithm a part of Calculus of Variations?

As in https://en.wikipedia.org/wiki/Calculus_of_variations The calculus of variations is a field of mathematical analysis that uses variations, which are small changes in functions and ...
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1answer
79 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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41 views

Where does the expectation term in the derivative of the soft-max policy come from?

At slide 17 of the David Silver's series, the soft-max policy is defined as follows $$ \pi_\theta(s, a) \propto e^{\phi(s, a)^T \theta} $$ that is, the probability of an action $a$ (in state $s$) is ...
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26 views

Since there are different types of neurons in adjacent positions in the brain's arrays, should heterogeneous layers be developed?

Below is a taxonomy of neurons. Some of these types occur in different locations in the brain, but there are adjacent neurons of varying types with clearly functional type diversity in many parts of ...
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2answers
231 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 ...
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0answers
55 views

Optimization step in Apprenticeship Learning via Inverse Reinforcement Learning

Why the optimization step of the algorithm a quadratic program? [See: Apprenticeship Learning via Inverse Reinforcement Learning; page 3] Isn't the objective function linear? Why don't we treat ...
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77 views

How does this sigma work?(Harris algorithm) [closed]

May someone explains some first iterations of this sigma? Also, how did it convert the above expression to below expression? What it the meaning of I(x) and I(y)?
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1answer
25 views

How to calculate the gradient penalty proposed in “Improved Training of Wasserstein GANs”?

The research paper titled Improved Training of Wasserstein GANs proposed a gradient penalty in order to avoid undesired behavior due to weight clipping of the discriminator. We now propose an ...
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2answers
36 views

What is meant by an axis of a tensor?

Tensor is an ordered collection of elements. The elements are generally real numbers. Tensors are used in deep learning for storing data. There is a wide usage of the word "axis" related to ...
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1answer
120 views

What is Bayes' theorem?

What is Bayes' theorem? How does it relate to conditional probabilities?
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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
33 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
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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1answer
83 views

Mathematical calculation behind decision tree classifier with continuous variables

I am working on a binary classification problem having continuous variables (Gene expression Values). My goal is to classify the samples as case or ...
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1answer
45 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....
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1answer
64 views

How to understand the average l2 loss?

In the snippet below, the highlighted part is the average norm, but since $1/|p_i|$ is outside the summation, it is very confusing to understand. is $|p_i|$ l2-norm(as per wolfram) or l1-norm or ...
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1answer
18 views

What is meant by “well-behaved gradient” in this context?

Consider the following statement about the success of discriminative models So far, the most striking successes in deep learning have involved discriminative models, usually those that map a high-...
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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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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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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
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

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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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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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
65 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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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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31 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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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 ...
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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 ...
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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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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 ...