Questions tagged [math]

For questions about mathematics related to artificial intelligence.

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2
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1answer
92 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
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1answer
68 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) + \...
6
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3answers
221 views

Is it ok to struggle with mathematics while learning AI as a beginner? [closed]

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 ...
3
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0answers
47 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 ...
2
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1answer
191 views

Is there a way of representing the minimax algorithm mathematically?

I have successfully figured out how the minimax algorithm works for a game like chess, where a game tree is used, and you assign a value to the terminal nodes and propagate that value up the tree. Is ...
3
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1answer
1k 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 ...
1
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1answer
703 views

What is the intuition behind the entropy formula used in the ID3 algorithm?

What is the intuition behind the following entropy formula used in the ID3 algorithm? $$ \text{info}(D) = -\sum_{i=1}^m p_i \log_2(p_i) $$
5
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1answer
471 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
61 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?
5
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1answer
144 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 ...
0
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1answer
39 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
58 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 ...
3
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2answers
89 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 ...
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2answers
53 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: ...
2
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1answer
144 views

Is there a mathematical example for Conditional Random Fields?

I am learning about probabilistic graphical models and I was wondering if there is an example explaining the math behind conditional random fields. Looking solely on the formula, I have no idea what ...
11
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2answers
4k views

Is the mean-squared error always convex in the context of neural networks?

Multiple resources I referred to mention that MSE is great because it's convex. But I don't get how, especially in the context of neural networks. Let's say we have the following: $X$: training ...
8
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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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1answer
85 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? ...
2
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1answer
183 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 ...
3
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1answer
444 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 ...
2
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1answer
47 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 $p(x)$?
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4answers
2k views

AI applications of the Fibonacci series

I have been looking at Fibonacci series, the golden ratio and its uses in nature, like how flowers and animals grow based on the series. I was wondering whether we could use the Fibonacci series and ...
4
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1answer
178 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 ...
5
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1answer
116 views

Is there a limit of minimum error for a particular training dataset in artificial Neural Network?

In error-based learning using gradient descent, if I give you a training dataset, then can you find the minimum error after training? And the minimum error should be true for all architectures of a ...
1
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1answer
76 views

What does “probabilistically” mean?

I'm reading the A. E. Eiben and J. E. Smith book Introduction to Evolutionary Computing (Springer 2003). On section 3.5 Recombination, page 47, the second paragraph said: Recombination operators ...
8
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2answers
2k views

Which areas of applied math are relevant to AI? [duplicate]

My background is in electrical engineering. I have a good grasp of CS foundations (e.g. data structures, algorithms, operating systems, discrete math and software engineering). I have option of ...
2
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1answer
55 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
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2answers
97 views

Are there any discount-factors based on branching factors?

I recently came across this function: $$\sum_{t = 0}^{\infty} \gamma^t R_t.$$ It's elegant and looks to be useful in the type of deterministic, perfect-information, finite models I'm working with. ...
3
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2answers
507 views

Which neural network should I use to approximate a specific function?

We have convolutional neural networks and recurrent neural networks for analysing respectively images and sequential data. How do I determine which neural network architecture is more appropriate to ...
5
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3answers
204 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
178 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 ...
3
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1answer
87 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 ...
4
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2answers
467 views

Viola Jones Algorithm

Can Viola Jones algorithm be used to detect the facial emotion. Actually it was used in creating harr-cascade file for object and facial detection, but what confused me is whether it can be used to ...
4
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0answers
113 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 ...
2
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0answers
49 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|...
6
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2answers
1k 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 ...
5
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2answers
114 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 ...
4
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1answer
95 views

Why does the cost function contain a 2 at the denominator?

A cost function used in machine is often the following $$C = \frac{1}{2} \| y - \hat{y} \| ^2$$ Why is there $\frac{1}{2}$ in front of the squared distance?
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3answers
92 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) =...
1
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1answer
66 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 ...
2
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0answers
31 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 $\...
5
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0answers
141 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 ...
1
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0answers
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 ...
10
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5answers
5k views

Why do activation functions need to be differentiable in the context of neural networks?

Why should an activation function of a neural network be differentiable? Is it strictly necessary or is it just advantageous?
1
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1answer
173 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 ...
29
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5answers
21k views

Is it possible to train the neural network to solve math equations?

I'm aware that neural networks are probably not designed to do that, however asking hypothetically, is it possible to train the deep neural network (or similar) to solve math equations? So given the ...
2
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1answer
288 views

Problems getting ADADELTA to converge

I have followed the pseudocode in the ADADELTA paper (top right on page 3), and wrote the following Python code for solving the optimization problem L(x) = x^2: ...
1
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0answers
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 ...
2
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1answer
700 views

Is known math really enough for AI

As an Electronics & Communication Engineering student I've heard some stories and theories about "The math we have is not enough to complete a thinker-learner AI." What is the truth? Is humankind ...
5
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1answer
551 views

Is recursion used in practice to improve performance of AI systems?

Is there any methods by which artificial intelligence use recursion(s) to solve a certain issue or to keep up working and calculating?