Questions tagged [math]
For questions about mathematics related to artificial intelligence.
82
questions
1
vote
1answer
22 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
80 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
42 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
51 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
184 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
110 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
75 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
35 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
26 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
$$
\operatorname{arcosh} \left( 1 + \frac{2|pq|^2|...
3
votes
1answer
67 views
Standard deviation of the total input to a neuron
Raul Rojas' Neural Networks A Systematic Introduction, section 8.2.1 calculates the standard deviation of the output of a hidden neuron.
From:
$$
\sigma^2 = \sum^n_{i=0}E[w_i^2]E[x_i^2]
$$
When I ...
6
votes
3answers
182 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
2answers
62 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
100 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$ ...
4
votes
3answers
90 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
35 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
25 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
68 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 ...
7
votes
1answer
525 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 ...
1
vote
0answers
12 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
178 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
1answer
47 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
21 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
270 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
54 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
137 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
51 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}$$
where $R_t$ is the reward at ...
0
votes
1answer
65 views
How are vectors and matrices multiplied in supervised machine learning?
I've recently started reading a book about deep learning. The book is titled "Grokking Deep Learning" (by Andrew W Trask). In chapter 3 (pages 44 and 45), it talks about multiplying vectors using dot ...
1
vote
0answers
27 views
Where does the expectation term in the derivative of the soft-max policy come from?
At slide 17 of the David Silver's series, the soft-max policy is defined as follows
$$
\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
52 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
138 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 ...
4
votes
1answer
227 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?
3
votes
3answers
200 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 ...
0
votes
1answer
55 views
Is the parent cost in A* added in every extended child?
How do we determine the cost of the parent path to its child in A* ("A star") search?
1
vote
0answers
192 views
Solving equations using reinforcement learning
I was lately curious about a reinforcement learning approach that would solve maths equations.
For example, if I have the following equation:
$$ f(g(h(w))) = 0 , with \ w = \begin{matrix}
a_{11} &...
5
votes
1answer
55 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
votes
2answers
212 views
What are the skills and disciplines I need to learn to get a job in Artificial Intelligence?
I'm in high school but hoping to have a career in artificial intelligence. What should I be pursuing educationally to get into this field?
1
vote
1answer
69 views
Reward-related formulation in reinforcement learning
I am referring to eq. 3.6 (p/g 49) based on Sutton's online book and can be found in an image below. I could not make sense of the final derivation of the equation $r(s, a, s')$. My question is ...
7
votes
3answers
528 views
Which areas of applied math are relevant to AI?
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 ...
3
votes
2answers
522 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 ...
1
vote
1answer
52 views
Weight Normalization paper
I am trying to dissect paper about weight normalization:
https://papers.nips.cc/paper/6114-weight-normalization-a-simple-reparameterization-to-accelerate-training-of-deep-neural-networks.pdf
...
5
votes
3answers
165 views
What are the algebraic properties of intelligence?
Some have said, "Two heads are better than one." That's true if they are collaborating. If not, the two together may be worse than zero.
Although that's a rhetorical opening paragraph, this is a ...
3
votes
2answers
467 views
Why is the derivative 0 if the policy is deterministic?
In the Berkeley RL class they mention the gradient would be 0 if the policy is deterministic. Why is that?
https://www.youtube.com/watch?v=XGmd3wcyDg8&feature=youtu.be&t=1071
5
votes
1answer
66 views
What makes multi-layer neural networks be able to perform nonlinear operations?
As I know, a single layer neural network can only do linear operations, but multilayered ones can.
Alao I recently learned that finite matrices/tensors, which are used in many neural networks can ...
3
votes
2answers
79 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.
...
1
vote
0answers
33 views
Optimization step in Apprenticeship Learning via Inverse Reinforcement Learning
Why the optimization step of the algorithm a quadratic program? [See: Apprenticeship Learning via Inverse Reinforcement Learning; page 3]
Isn't the objective function linear? Why don't we treat ...
