Questions tagged [math]

For questions about mathematics related to artificial intelligence.

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17 views

Is the derivative of the loss wrt a single scalar parameter proportional to the loss?

I am wondering about the correlation between the loss and the derivative of the loss wrt a single scalar parameter, with the same sample. That means: considering a machine learning model with ...
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6 views

How do I decide which norm to use for placing a constraint on my adversarial perturbation?

I am performing an adversarial machine learning attack on a neural network for network traffic classification. For adding adversarial perturbations in features such as packet interarrival times and ...
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0answers
18 views

Is there any wrong in my focal loss derivation?

Assume $\mathbf{X} \in R^{N, C}$ is the input of the softmax $\mathbf{P} \in R^{N, C}$, where $N$ is number of examples and $C$ is number of classes: $$\mathbf{p}_i = \left[ \frac{e^{x_{ik}}}{\sum_{j=...
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1answer
28 views

Is ReLU a non-linear activation function?

According to this blog post The purpose of an activation function is to add some kind of non-linear property to the function The sigmoid is typically used as an activation function of a unit of a ...
2
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1answer
56 views

Why does variational auto-encoder use the reconstruction loss?

VAE is trained to reduce the following two losses. KL divergence between inferred latent distribution and Gaussian. the reconstruction loss I understand that the first one regularizes VAE to get ...
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0answers
25 views

Why are conics important in computer vision?

The book Multiple View Geometry in Computer Vision by Richard Hartley and Andrew Zisserman talks about lines, points and conics. A conic is a curve described by a second-degree equation in the plane, ...
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34 views

Derivation of regularized cost function w.r.t activation and bias

In regularzied cost function a L2 regularization cost has been added. Here we have already calculated cross entropy cost w.r.t $A, W$. As mentioned in the regularization notebook (see below) in ...
2
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1answer
57 views

Why is my derivation of the back-propagation equations inconsistent with Andrew Ng's slides from Coursera?

I am using the cross-entropy cost function to calculate its derivatives using different variables $Z, W$ and $b$ at different instances. Please refer image below for calculation. As per my knowledge, ...
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1answer
38 views

In a single neuron output layer should the output be a scalar?

Given a neural network with 3 inputs, 4 hidden layers, and 1 output, should the output neuron be a vector or a scalar? I thought that at the end of the summation only one number between 0 and 1 would ...
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0answers
48 views

Interpretation of inverse matrix in mean calculation in Gaussian Process

The formula for mean prediction using Gaussian Process is $k(x_*, x)k(x, x)^{-1}y$, where $k$ is the covariance function. See e.g. equation 2.23 (in chapter 2) from Gaussian Processes for Machine ...
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1answer
104 views

Understanding the derivation of the first-order model-agnostic meta-learning

According to the authors of this paper, to improve the performance, they decided to drop backward pass and using a first-order approximation I found a blog which discussed how to derive the math ...
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2answers
83 views

Is there a possibility that there is no relationship between some inputs and outputs?

I'm doing machine learning projects. I took a look at many datasets I worked with, mostly there are already famous datasets that everyone uses. Let's say I decided to make my own dataset. Is there a ...
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30 views

Expected duration in a state

I am going through Rabiner 1989 and he writes that the discrete probability density function of duration $d$ in state $i$ (that is, staying in a state for duration $d$, conditioned on starting in that ...
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1answer
31 views

What are the variables used in a Gaussian radial basis kernel in the context of SVMs?

If I have the Gaussian kernel $$ k(x, x') = \operatorname{exp}\left( -\| x - x' \|^2 / 2\sigma^2 \right) $$ What is $x$ and $x'$ in the context of training an SVM?
2
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1answer
47 views

How is the expected value in the loss function of DQN approximated?

In Deep Q Learning the parametrized Q-functions $Q_i$ are optimised by performing gradient descent on the series of loss functions $L_i(\theta_i)= E_{(s,a)\sim p}[(y_i-Q(s,a;\theta_i))^2]$ , where ...
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2answers
31 views

What are the conditions for the convergence of SARSA to the optimal value function?

Is it correct that for SARSA to converge to the optimal value function (and policy) The learning rate parameter $\alpha$ must satisfy the conditions: $$\sum \alpha_{n^k(s,a)} =\infty \quad \text{and}...
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1answer
21 views

What is the equation of the separation line for this neuron with identity activation?

I have a single neuron with 2 inputs, and identity activation, where f is activation function and u is output: $u = f(w_1x_1 + ...
1
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1answer
45 views

Does TD(0) prediction require Robbins-Monro conditions to converge to the value function?

Does the learning rate parameter $\alpha$ require the Robbins-Monro conditions below for the TD(0) algorithm to converge to the true value function of a policy? $$\sum \alpha_t =\infty \quad \text{...
3
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1answer
44 views

In the policy gradient equation, is $\pi(a_{t} | s_{t}, \theta)$ a distribution or a function?

I am learning about policy gradient methods from the Deep RL Bootcamp by Peter Abbeel and I am a bit stumbled by the math presented. In the lecture, he derives the gradient logarithm likelihood of a ...
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1answer
43 views

Formal proof that every purely reactive agent has behaviorally equivalent standard agent

It kind of makes sense intuitively but I'm not sure about a formal proof. I'll start with briefly listing definitions from Intro to Multiagent systems, Wooldridge, 2002 and then give you my reasoning ...
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0answers
38 views

Understanding V- and Q-functions

Assume the existence of a Markov Decision Process consisting of: State space $S$ Action space $A$ Transition model $T: S \times A \times S \to [0,1]$ Reward function $R: S \times A \times S \to \...
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2answers
105 views

Mathematical foundations of the ability to learn

I am an undergraduate student in applied mathematics with an interest in artificial intelligence. I am currently exploring topics where I could do research. Coming from a mathematical background I am ...
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1answer
36 views

Why is Standard Deviation based on L2 Variance and not L1 Variance

Standard deviation and variance are in statistics but the formula for variance is somehow related to the L1 and L2. Mathematically (L2 in machine learning sense), $$Variance = \dfrac{(X_1-Mean)^2+..+(...
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0answers
29 views

Is Gradient Descent algorithm a part of Calculus of Variations?

As in https://en.wikipedia.org/wiki/Calculus_of_variations The calculus of variations is a field of mathematical analysis that uses variations, which are small changes in functions and ...
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1answer
33 views

How to understand the average l2 loss?

In the snippet below, the highlighted part is the average norm, but since $1/|p_i|$ is outside the summation, it is very confusing to understand. is $|p_i|$ l2-norm(as per wolfram) or l1-norm or ...
3
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1answer
80 views

What is the difference between the notations $\|x\|_1, \|x\|_2$ and $|x|$?

What is the difference between the notations $\|x\|_1, \|x\|_2$ and $|x|$? I think $|x|$ is the magnitude of $x$.
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0answers
50 views

How does the update rule for the one-step actor-critic method work?

Can you please elucidate the math behind the update rule for the critic? I've seen in other places that just a squared distance of $R + \hat{v}(S', w) - \hat{v}(S,w)$ is used, but Sutton suggests an ...
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2answers
158 views

Can we get the inverse of the function that a neural network represents?

I was wondering if it's possible to get the inverse of a neural network. If we view a NN as a function, can we obtain its inverse? I tried to build a simple MNIST architecture, with the input of (784,...
5
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1answer
228 views

What is the mathematical definition of an activation function?

What is the mathematical definition of an activation function to be used in a neural network? So far I did not find a precise one, summarizing which criterions (e.g. monotonicity, differentiability, ...
4
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0answers
58 views

Is there a mathematical formula that describes the learning curve in neural networks?

In training a neural network, you often see the curve showing how fast the neural network is getting better. It usually grows very fast then slows down to almost horizontal. Is there a mathematical ...
2
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1answer
73 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
48 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) + \...
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0answers
41 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
137 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), ...
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2answers
177 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
375 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
241 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
57 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?
3
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1answer
41 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
49 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 ...
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1answer
48 views

Trying to understand the math of GANs with Goodfellows paper and tutorial

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}_{\...
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1answer
29 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
42 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 ...
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2answers
26 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: ...
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3answers
147 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
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1answer
51 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 ...
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1answer
40 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 $\theta$?...
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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
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3answers
199 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
36 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 $p(x)$?