Questions tagged [math]

For questions about mathematics related to artificial intelligence.

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3
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1answer
58 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-...
2
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1answer
58 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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0answers
28 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 ...
2
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1answer
48 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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0answers
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 ...
4
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4answers
458 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 ...
0
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0answers
38 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 ...
4
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1answer
72 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 ...
2
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1answer
44 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
54 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
31 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
30 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\}_{...
1
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1answer
57 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 ...
0
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1answer
54 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 ...
2
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1answer
144 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 ...
2
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1answer
92 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 ...
6
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1answer
133 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 ...
3
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0answers
20 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
43 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 ...
1
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1answer
69 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 ...
0
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1answer
51 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 ...
2
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2answers
114 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{...
1
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1answer
49 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
56 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 ...
3
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1answer
123 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 ...
1
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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
44 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 ...
1
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0answers
110 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
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 ...
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 ...
2
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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
119 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
89 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
78 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
68 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. ...
0
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1answer
43 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....
1
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1answer
96 views

Understanding relation between VC Symmetrization Lemma and Generalization Bounds

I am new in the field of Machine Learning so I wanted to start of by reading more about mathematics and history behind it. I am currently reading, in my opinion, a very good and descriptive paper on ...
2
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1answer
50 views

How can I implement policy evaluation when reward is tied to an action outcome?

I'm following Stanford reinforcement learning videos on youtube. One of the assignments asks to write code for policy evaluation for Gym's FrozenLake-v0 environment. In the course (and books I have ...
2
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1answer
80 views

What are the challenges faced by using NLP to convert mathematical texts into formal logic?

From what I've figured (a) converting mathematical theorems and proofs from English to formal logic is a straightforward job for mathematicians with sufficient background, except that it takes time. ...
1
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1answer
53 views

What does equation in the “related work” section of the GAN paper mean?

I was going through the paper on GAN by Ian Goodfellow. Under the related work section, there is an equation. I cannot decipher the equation. Can anyone help me understand the meaning of the equation? ...
3
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0answers
42 views

Is maximum likelihood estimation meaningless for a dataset of only outliers?

From my understanding, maximum likelihood estimation chooses the set of parameters for the estimator that maximizes likelihood with the ground truth distribution. I always interpreted it as the ...
2
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1answer
47 views

Is the derivative of the loss wrt a single scalar parameter proportional to the loss?

I am wondering about the correlation between the loss and the derivative of the loss wrt a single scalar parameter, with the same sample. That means: considering a machine learning model with ...
1
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0answers
19 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 ...
2
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0answers
28 views

Is there any wrong in my focal loss derivation?

Assume $\mathbf{X} \in R^{N, C}$ is the input of the softmax $\mathbf{P} \in R^{N, C}$, where $N$ is number of examples and $C$ is number of classes: $$\mathbf{p}_i = \left[ \frac{e^{x_{ik}}}{\sum_{j=...
2
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2answers
121 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 ...
3
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1answer
123 views

Why does the variational auto-encoder use the reconstruction loss?

VAE is trained to reduce the following two losses. KL divergence between inferred latent distribution and Gaussian. the reconstruction loss I understand that the first one regularizes VAE to get ...