Questions tagged [math]

For questions about mathematics related to artificial intelligence.

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2
votes
1answer
81 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
59 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
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0answers
45 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
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1answer
152 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
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2answers
298 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
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1answer
548 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
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1answer
275 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
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1answer
58 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?
4
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1answer
72 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
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2answers
54 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
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1answer
58 views

Why does the discriminator minimize the cross-entropy while the generator maximize it?

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
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1answer
33 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
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1answer
46 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
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2answers
34 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
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3answers
230 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 ...
3
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1answer
75 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
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1answer
45 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 $\theta$?...
7
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3answers
2k 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
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3answers
209 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
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1answer
39 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
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0answers
59 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
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2answers
81 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
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1answer
81 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
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1answer
88 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....
11
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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
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1answer
49 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 ...
4
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1answer
147 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
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3answers
195 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
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1answer
154 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
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1answer
74 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
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0answers
39 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
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1answer
79 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
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2answers
534 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
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3answers
104 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
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2answers
110 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
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1answer
69 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
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2answers
68 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
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0answers
86 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 ...
12
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1answer
3k 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 a function where the ...
1
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0answers
21 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
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2answers
889 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
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2answers
172 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^{...
1
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3answers
89 views

Is it possible to compute $P( F \mid S )$ given $P(F \mid S,A)$, $P(F \mid S, \lnot A)$?

I have a bayesian network, which has the following data: $P(S) = 0.07$ $P(A) = 0.01$ $P(F \mid S,A) = 1.0$ $P(F \mid S, \lnot A) = 0.7$ $P(F \mid \lnot S, A) = 0.9$ $P(F \mid \lnot S, \lnot A) =...
2
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0answers
28 views

Calculating tangent vector of curve s(P,$\alpha$) at given point $\alpha$ = 0

I am reading the paper "Transformation Invariance in Pattern Recognition – Tangent Distance and Tangent Propagation", where the tangent vector is calculated for the given curve $s(P,\alpha)$ at $\...
7
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2answers
1k views

Why exactly do neural networks require i.i.d. data?

In reinforcement learning, in general, successive states (actions and rewards) are highly correlated. An "experience replay" buffer was used, in the DQN architecture, to avoid training the neural ...
4
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0answers
83 views

What are the main benefits of using Bayesian networks?

I have some trouble understanding the benefits of Bayesian networks. Am I correct that the key benefit of the network is that one does not need to use chain rule of probability in order to calculate ...
3
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4answers
311 views

Can we define the AI singularity mathematically?

The "AI Singularity" or "Technological Singularity" is a vague term that roughly seems to refer to the idea of: Humans can design algorithms Humans can improve algorithms Eventually algorithms we ...
1
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1answer
63 views

Is the next state drawn from the joint distribution of the previous state and action?

Suppose $G_t$, the discounted return at time $t$ is defined as: $$ G_t \triangleq R_t+\gamma R_{t+1}+\gamma^{2}R_{t+2} + \cdots = \sum_{j=1}^{\infty} \gamma^{k}R_{t+k}$$ where $R_t$ is the reward at ...
1
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1answer
158 views

How are vectors and matrices multiplied in supervised machine learning?

I've recently started reading a book about deep learning. The book is titled "Grokking Deep Learning" (by Andrew W Trask). In chapter 3 (pages 44 and 45), it talks about multiplying vectors using dot ...
1
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0answers
36 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 ...