# 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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### 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 ...
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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### 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 ...
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 ...
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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### 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 ...
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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### 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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### 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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### 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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### 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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### 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 ...
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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### 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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### 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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### 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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### 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 ...
120 views

### What is Bayes' theorem?

What is Bayes' theorem? How does it relate to conditional probabilities?
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### 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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### 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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### 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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### 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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### 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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### 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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### 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-...
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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### 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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### (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, ...
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: ...
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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### 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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### 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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### 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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### 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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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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### 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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### 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 ...