Questions tagged [math]
For questions about mathematics related to artificial intelligence.
97
questions
3
votes
1answer
31 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
votes
0answers
28 views
Show that if $H$ is PAC learnable (in the standard one-oracle model), then $H$ is also learnable in the two-oracle model
Consider a variant of the PAC model in which there are two example oracles: one that generates positive examples and one that generates negative examples, both according to the underlying distribution ...
0
votes
2answers
22 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
votes
2answers
85 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
38 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
33 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
157 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 ...
1
vote
1answer
27 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
47 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
43 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
50 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
107 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 ...
8
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
43 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
72 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
191 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
115 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
103 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
40 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
29 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
$$
\operatorname{arcosh} \left( 1 + \frac{2|pq|^2|...
3
votes
1answer
71 views
Standard deviation of the total input to a neuron
Raul Rojas' Neural Networks A Systematic Introduction, section 8.2.1 calculates the standard deviation of the output of a hidden neuron.
From:
$$
\sigma^2 = \sum^n_{i=0}E[w_i^2]E[x_i^2]
$$
When I ...
6
votes
3answers
205 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 ...
1
vote
3answers
74 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
103 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
58 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
54 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 ...
4
votes
3answers
108 views
How can a collaboration game be defined mathematically?
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
38 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
27 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
69 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
853 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
15 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
265 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., ...
1
vote
2answers
64 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
vote
3answers
87 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
votes
0answers
23 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 $\...
4
votes
2answers
386 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
votes
0answers
62 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 ...
2
votes
2answers
151 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
vote
1answer
55 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}$$
where $R_t$ is the reward at ...
0
votes
1answer
73 views
How are vectors and matrices multiplied in supervised machine learning?
I've recently started reading a book about deep learning. The book is titled "Grokking Deep Learning" (by Andrew W Trask). In chapter 3 (pages 44 and 45), it talks about multiplying vectors using dot ...
1
vote
0answers
28 views
Where does the expectation term in the derivative of the soft-max policy come from?
At slide 17 of the David Silver's series, the soft-max policy is defined as follows
$$
\pi_\theta(s, a) \propto e^{\phi(s, a)^T \theta}
$$
that is, the probability of an action $a$ (in state $s$) is ...
1
vote
0answers
16 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
votes
1answer
59 views
Choice of fuzzification function
I'm a relative newbie to fuzzie logic systems but I have some knowledge in mathematics. I have the following problem:
I want to fuzzify certain values. Some are in the range [-$\inf$,$\inf$] and some ...
3
votes
1answer
226 views
Why does the “reward to go” trick in policy gradient methods work?
In policy gradient method, there's a trick to reduce a variance of policy gradient. We use causality, and remove part of the sum over rewards so that only actions happened after the reward are taken ...
1
vote
1answer
46 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 ...
6
votes
1answer
311 views
How do we prove the n-step return error reduction property?
In section 7.1 (about the n-step bootstrapping) of the book Reinforcement Learning: An Introduction (2nd edition), by Andrew Barto and Richard S. Sutton, the authors write about what they call the "n-...
-1
votes
1answer
41 views
When will we have computer programs that can compose mathematical proofs?
When will it be possible to give a computer program a bunch of assumptions and ask it if a certain statement is true or false, giving a proof or a counterexample respectively?
