Questions tagged [math]
For questions about mathematics related to artificial intelligence.
137
questions
1
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0answers
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 ...
0
votes
0answers
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 ...
2
votes
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=...
0
votes
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
votes
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 ...
1
vote
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, ...
0
votes
0answers
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
votes
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, ...
1
vote
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 ...
2
votes
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 ...
1
vote
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 ...
2
votes
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 ...
2
votes
0answers
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 ...
1
vote
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)
$$
What is $x$ and $x'$ in the context of training an SVM?
2
votes
1answer
47 views
How is the expected value in the loss function of DQN approximated?
In Deep Q Learning the parametrized Q-functions $Q_i$ are optimised by performing gradient descent on the series of loss functions
$L_i(\theta_i)= E_{(s,a)\sim p}[(y_i-Q(s,a;\theta_i))^2]$ , where
...
1
vote
2answers
31 views
What are the conditions for the convergence of SARSA to the optimal value function?
Is it correct that for SARSA to converge to the optimal value function (and policy)
The learning rate parameter $\alpha$ must satisfy the conditions:
$$\sum \alpha_{n^k(s,a)} =\infty \quad \text{and}...
1
vote
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
vote
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?
$$\sum \alpha_t =\infty \quad \text{...
3
votes
1answer
44 views
In the policy gradient equation, is $\pi(a_{t} | s_{t}, \theta)$ a distribution or a function?
I am learning about policy gradient methods from the Deep RL Bootcamp by Peter Abbeel and I am a bit stumbled by the math presented. In the lecture, he derives the gradient logarithm likelihood of a ...
1
vote
1answer
43 views
Formal proof that every purely reactive agent has behaviorally equivalent standard agent
It kind of makes sense intuitively but I'm not sure about a formal proof. I'll start with briefly listing definitions from Intro to Multiagent systems, Wooldridge, 2002 and then give you my reasoning ...
0
votes
0answers
38 views
Understanding V- and Q-functions
Assume the existence of a Markov Decision Process consisting of:
State space $S$
Action space $A$
Transition model $T: S \times A \times S \to [0,1]$
Reward function $R: S \times A \times S \to \...
4
votes
2answers
105 views
Mathematical foundations of the ability to learn
I am an undergraduate student in applied mathematics with an interest in artificial intelligence. I am currently exploring topics where I could do research. Coming from a mathematical background I am ...
1
vote
1answer
36 views
Why is Standard Deviation based on L2 Variance and not L1 Variance
Standard deviation and variance are in statistics but the formula for variance is somehow related to the L1 and L2.
Mathematically (L2 in machine learning sense),
$$Variance = \dfrac{(X_1-Mean)^2+..+(...
1
vote
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 ...
0
votes
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
votes
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 $x$.
2
votes
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 ...
6
votes
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
votes
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
votes
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
votes
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
votes
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) + \...
3
votes
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
votes
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), ...
2
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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 ...
1
vote
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}_{\...
0
votes
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
votes
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 ...
0
votes
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:
...
4
votes
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
votes
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 ...
1
vote
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 $\theta$?...
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
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
votes
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 $p(x)$?
