Questions tagged [math]

For questions about mathematics related to artificial intelligence.

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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
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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
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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
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
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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
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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
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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
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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 (...
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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
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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
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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 ...
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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
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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
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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
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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
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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
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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, ...
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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
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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
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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 ...
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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 ...
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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., ...
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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^{...
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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) =...
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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
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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
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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
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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
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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 ...
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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 ...
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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 ...
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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
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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
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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
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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
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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
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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
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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?
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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 ...