Questions tagged [math]
For questions about mathematics related to artificial intelligence.
113
questions
3
votes
1answer
65 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,...
5
votes
1answer
201 views
What is the mathematical definition of an activation function?
What is the mathematical definition of an activation function to be used in a neural network?
So far I did not find a precise one, summarizing which criterions (e.g. monotonicity, differentiability, ...
4
votes
0answers
55 views
Is there a mathematical formula that describes the learning curve in neural networks?
In training a neural network, you often see the curve showing how fast the neural network is getting better. It usually grows very fast then slows down to almost horizontal.
Is there a mathematical ...
2
votes
1answer
68 views
What is the neuron-level math behind backpropagation for a neural network?
I am quite new in the AI field. I am trying to create a neural network, in a language (Dart) where I couldn't find examples or premade libraries or tutorials. I've tried looking online for a strictly "...
3
votes
1answer
42 views
What does the notation sup dist mean in distributional RL?
I'm trying to understand distributional RL, based on this article. In one of the equations, there is a symbol $\operatorname{sup dist}$.
\begin{align}
\operatorname{sup dist}_{s, a} (R(s, a) + \...
3
votes
0answers
37 views
How does the memory mechanism (reading and writing) work in a neural Turing machine?
In neural Turing machine (NTM), reading memory is represented as
\begin{align}
r_t \leftarrow \sum\limits_i^R w_t(i) \mathcal{M}_t(i) \tag{2}
\end{align}
and writing to memory is represented as
...
8
votes
1answer
127 views
How does the forget layer of an LSTM work?
Can someone explain the mathematical intuition behind the forget layer of an LSTM?
So as far as I understand it, the cell state is essentially long term memory embedding (correct me if I'm wrong), ...
2
votes
2answers
130 views
What is probability distribution in machine learning?
If we were learning or working in machine learning field then we frequently come across this term probability distribution. I know what probability, conditional probability and probability ...
3
votes
1answer
243 views
What does the Markov assumption say about the history of state sequences?
Does the Markov assumption say that the conditional probability of the next state only depends on the current state or does it say that the conditional probability depends on a fixed finite number of ...
5
votes
1answer
209 views
What is “conditioning” on a feature?
On page 98 of Jet Substructure at the Large Hadron Collider: A Review of Recent Advances in Theory and Machine Learning the author writes;
Redacted phase space: Studying the distribution of inputs ...
4
votes
1answer
57 views
What do the subscripts mean in $N_{t,n,\sigma,L}$?
A neural network can apparently be denoted as $N_{t,n,\sigma,L}$. What do these subscripts $t, n, \sigma$ and $L$ mean? Could you link me to a paper, article or webpage with an explanation for this?
3
votes
1answer
28 views
Why does a Lipschitz continuous discriminator in GANs assure statistical boundedness?
I have been reading the paper which introduced spectral normalization in GANs.
At some point the paper mentions the following:
The machine learning community has been pointing out recently that ...
4
votes
2answers
48 views
Which linear algebra book should I read to understand vectorized operations?
I am reading the Goodfellow's book about neural networks, but I am stuck in the mathematical calculus of the back-propagation algorithm. I understood the principle, and some Youtube videos explaining ...
1
vote
1answer
47 views
Trying to understand the math of GANs with Goodfellows paper and tutorial
In his original GAN paper Goodfellow gives a game theoretic perspective for GANs:
\begin{equation}
\underset{G}{\min}\, \underset{D}{\max}\, V\left(D,G \right) =
\mathbb{E}_{x\sim\mathit{p}_{\...
0
votes
1answer
28 views
Backpropagation: Chain Rule to the Third Last Layer
I'm trying to solve dLoss/dW1. The network is as in picture below with identity activation at all neurons:
Solving dLoss/dW7 is simple as there's only 1 way to output:
$Delta = Out-Y$
$Loss = abs(...
3
votes
1answer
38 views
What are the differences between stability and convergence in reinforcement learning?
The terms are mentioned in the paper: “An Emphatic Approach to the Problem of off-Policy Temporal-Difference Learning.” (Sutton, Mahmood, White; 2016) and more, of course.
In which paper, they ...
0
votes
2answers
24 views
Is it still called linear separation with a layer of more than 1 neuron
A single neuron will be able to do linear separation. For example, XOR simulator network:
...
4
votes
3answers
126 views
How can I determine the mathematical relation between the input and output variables?
I would like to take in some input values for $n$ variables, say $R$, $B$, and $G$. Let $Y$ denote the response variable of these $n$ inputs (in this example, we have $3$ inputs). Other than these, I ...
2
votes
1answer
40 views
How can I derive the rotation matrix from the axis-angle rotation vector?
Given an axis-angle rotation vector $\Theta = (2,2,0)$, after finding the unit vector $k=(1/\sqrt{2}, 1/\sqrt{2}, 0)$ and angle $\theta = 2\sqrt{2}$ representing the same rotation, I need to derive ...
1
vote
1answer
40 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 $\theta$?...
6
votes
3answers
1k views
Why is the derivative of the activation functions in neural networks important?
I'm new to NN. I am trying to understand some of its foundations. One question that I have is: why the derivative of an activation function is important (not the function itself), and why it's the ...
4
votes
3answers
194 views
Is it ok to struggle with mathematics while learning AI as a beginner?
I have a decent background in Mathematics and Computer Science .I started learning AI from Andrew Ng's course from one month back. I understand logic and intuition behind everything taught but if ...
2
votes
1answer
34 views
Why is the expectation calculated over finite number of points drawn from a probability distribution?
This is from the book Pattern Recognition by Bishop. Why is expectation here a simple average? Why is $f(x)$ not being multiplied by $p(x)$?
2
votes
0answers
49 views
What is the meaning of the words 'bias' and 'variance' in RL?
In algorithms like MC/TD (tabular value approximation) two of the metrics used to measure their performance are the bias and the variance. What do these terms mean? And which characteristic of the ...
4
votes
2answers
76 views
How would an AI work out this question?
I am trying to create an AI that makes reasonable guesses at truths of statements. However...
Human: "Prove that no number exists which is one more than a billion."
AI: "Is it true for the number 1? ...
1
vote
1answer
49 views
Is there a rigorous proof for finding Hopfield minima?
I am looking for a rigorous mathematical proof for finding the several local minima of the Hopfield networks. I am searching for something rigorous, a demonstration, not just let the network keep ...
2
votes
1answer
51 views
How can I learn tensors for deep learning?
I've seen in most deep learning papers use tensors. I understood what tensors are, but I want to dive into them, because I think that might be beneficial for further studies in Artificial Intelligence....
4
votes
1answer
253 views
Where can I find the proof of the universal approximation theorem?
The Wikipedia article for the universal approximation theorem cites a version of the universal approximation theorem for Lebesgue-measurable functions from this conference paper. However, the paper ...
10
votes
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 ...
2
votes
1answer
47 views
Is the Markov property assumed in the forward algorithm?
I'm majoring in pure linguistics (not computational), and I don't have any basic knowledge regarding computational science or mathematics. But I happen to take "Automatic Speech Recognition" course in ...
3
votes
1answer
109 views
How is G(z) related to x in GAN proof?
In the proofs for the original GAN paper, it is written:
$$∫_x p_{data}(x) \log D(x)dx+∫_zp(z)\log(1−D(G(z)))dz
=∫_xp_{data}(x)\log D(x)+p_G(x) \log(1−D(x))dx$$
I've seen some explanations asserting ...
4
votes
3answers
194 views
Which function $(\hat{y} - y)^2$ or $(y - \hat{y})^2$ should I use to compute the gradient?
The MSE can be defined as $(\hat{y} - y)^2$, which should be equal to $(y - \hat{y})^2$, but I think their derivative is different, so I am confused of what derivative will I use for computing my ...
4
votes
1answer
122 views
How are filters weights updated for a CNN?
I've been trying to learn backpropagation for CNNs. I read several articles like this one and this one. They all say that to compute the gradients for the filters, you just do a convolution with the ...
1
vote
1answer
245 views
Which machine learning models are universal function approximators?
The universal approximation theorem states that a feed-forward neural network with a single hidden layer containing a finite number of neurons can approximate a wide variety of interesting (...
1
vote
0answers
43 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 ...
2
votes
0answers
36 views
Should I use the hyperbolic distance loss in the case of Poincarè Disk Model?
I trained a neural network which makes a regression to a Poincarè Disk Model with radius $r = 1$.
I want to optimize using the hyperbolic distance
$$
\operatorname{arcosh} \left( 1 + \frac{2|pq|^2|...
3
votes
1answer
75 views
Standard deviation of the total input to a neuron
Raul Rojas' Neural Networks A Systematic Introduction, section 8.2.1 calculates the standard deviation of the output of a hidden neuron.
From:
$$
\sigma^2 = \sum^n_{i=0}E[w_i^2]E[x_i^2]
$$
When I ...
6
votes
2answers
322 views
How is local minima possible in gradient descent?
Gradient descent works on the equation of mean squared error, which is an equation of a parabola $y=x^2$.
We often say that weight adjustment in a neural network by gradient descent algorithm can hit ...
2
votes
3answers
97 views
What does the formula $1-\sum_i(e_i-a_i)^2$ mean in this NEAT Python API?
I have looked at the documentation for the NEAT Python API found here, but it shows calculus like this:
The error for each genome is $1-\sum_i(e_i-a_i)^2$
I haven't learned calculus at the moment....
5
votes
2answers
105 views
Are on-line backpropagation iterations perpendicular to the constraint?
Raul Rojas' Neural Networks A Systematic Introduction, section 8.1.2 relates off-line backpropagation and on-line backpropagation with Gauss-Jacobi and Gauss-Seidel methods for finding the ...
2
votes
1answer
61 views
Which matrix represents the similarity between words when using SVD?
Two words can be similar if they co-occur "a lot" together. They can also be similar if they have similar vectors. This similarity can be captured using cosine similarity. Let $A$ be a $n \times n$ ...
3
votes
2answers
57 views
Why is the max a non-expansive operator?
In certain reinforcement learning (RL) proofs, the operators involved are assumed to be non-expansive. For example, on page 6 of the paper Generalized Markov Decision Processes: Dynamic-programming ...
3
votes
3answers
119 views
How can a collaboration game be defined mathematically? [closed]
One of the common conceptions in AI is the idea of game theory. We see that in the predominance of chess and other games in the literature as metrics of AI success. We see it in the names of machine ...
2
votes
0answers
44 views
How can the convolution operation be implemented as a matrix-vector multiplication?
How can the convolution operation used by CNNs be implemented as a matrix-vector multiplication? We often think of the convolution operation in CNNs as a kernel that slides across the input. However, ...
0
votes
0answers
37 views
How is the convolution operation used in CNNs a special case of the convolution operator?
How is the convolution operation used in convolutional neural networks (CNNs) a special case of the mathematical convolution operator? Most of us, when we think of the "convolution operation", we ...
3
votes
0answers
73 views
What characteristics make it difficult for a Neural Network to approximate a function?
What are the characteristics which make a function difficult for the Neural Network to approximate?
Intuitively, one might think uneven functions might be difficult to approximate, but uneven ...
8
votes
1answer
2k views
What is the Bellman operator in reinforcement learning?
In mathematics, the word operator can refer to several distinct but related concepts. An operator can be defined as a function between two vector spaces, it can be defined as function where the domain ...
1
vote
0answers
16 views
Does a mechanical system repeats itself?
The inverted pendulum problem is a famous control task. It can be solved with a technique called system identification. System identification means to formalize the state-action space of a system in a ...
0
votes
2answers
530 views
What are examples of applications of the Fourier transform to AI?
The (discrete and continuous) Fourier transform (FT) is used in signal processing in order to convert a signal (or function) in a certain domain (e.g. the time domain) to another domain (e.g., ...
2
votes
2answers
114 views
Why is MSE used over other quadratic loss functions?
So I was wondering, why I have only encountered square loss function also known as MSE. The only nice property of MSE I am so far aware of is its convex nature. But then all equations of the form $x^{...
