Skip to main content

Unveiling Hidden Neural Codes: SIMPL – A Scalable and Fast Approach for Optimizing Latent Variables and Tuning Curves in Neural Population Data

Dr. Rishabh Pathak
This research paper presents SIMPL (Scalable Iterative Maximization of Population-coded Latents), a novel, computationally efficient algorithm designed to refine the estimation of latent variables and tuning curves from neural population activity. Latent variables in neural data represent essential low-dimensional quantities encoding behavioral or cognitive states, which neuroscientists seek to identify to understand brain computations better. Background and Motivation Traditional approaches commonly assume the observed behavioral variable as the latent neural code. However, this assumption can lead to inaccuracies because neural activity sometimes encodes internal cognitive states differing subtly from observable behavior (e.g., anticipation, mental simulation). Existing latent variable models face challenges such as high computational cost, poor scalability to large datasets, limited expressiveness of tuning models, or difficulties interpreting complex neural network-based functio...
Post a Comment

Logistic Regression

Image
Dr. Rishabh Pathak


Logistic regression is a fundamental classification algorithm widely used for binary and multi-class classification problems. 

1. What is Logistic Regression?

Logistic regression is a supervised learning algorithm designed for classification tasks, especially binary classification where the response variable y takes values in {0,1}. Unlike linear regression, which predicts continuous outputs, logistic regression predicts probabilities that an input x belongs to the positive class (y=1).

2. Hypothesis Function and Model Formulation

In logistic regression, the hypothesis function hθ​(x) models the probability p(y=1∣x;θ) using the logistic (sigmoid) function applied to a linear combination of input features:

hθ​(x)=P(y=1∣x;θ)=1+e−θTx1​

where:

  • θ∈Rd+1 are the parameters (weights),
  • x∈Rd+1 is the augmented feature vector (usually including a bias term),
  • θTx is the linear predictor,
  • the function g(z)=1+e−z1​ is the logistic or sigmoid function,.

This design ensures the output is always between 0 and 1, which can be interpreted as a probability.

3. Statistical Model and Bernoulli Distribution

Logistic regression assumes that the conditional distribution of y given x follows a Bernoulli distribution parameterized by ϕ=hθ​(x):

y∣x;θ∼Bernoulli(hθ​(x))

The expectation of y is:

E[y∣x;θ]=ϕ=hθ​(x)

The use of the Bernoulli distribution leads naturally to the logistic function through the generalized linear model (GLM) framework and the exponential family of distributions.

  • The canonical response function for Bernoulli is logistic sigmoid g(η)=1+e−η1​,
  • The canonical link function is the inverse of the response function g−1.

4. Parameter Estimation via Maximum Likelihood

Parameters θ are typically estimated by maximizing the likelihood of the observed data, or equivalently, minimizing the negative log-likelihood (also called the cross-entropy loss function). For training examples {(x(i),y(i))}i=1n​, the loss for a single example is:

J(i)(θ)=−logp(y(i)∣x(i);θ)=−(y(i)loghθ​(x(i))+(1−y(i))log(1−hθ​(x(i))))

And the total cost function is the average loss over all examples:

J(θ)=n1​∑i=1n​J(i)(θ)

The optimization is usually done using gradient descent or variants.

5. Multi-class Logistic Regression (Softmax Regression)

For multi-class classification where y∈{1,2,…,k}, logistic regression generalizes to the softmax function, mapping the outputs to a probability distribution over k classes:

Let the model outputs be logits hˉθ​(x)∈Rk, where each component corresponds to a class:

P(y=j∣x;θ)=∑s=1k​exp(hˉθ​(x)s​)exp(hˉθ​(x)j​)​

The loss function per training example is then the negative log likelihood:

J(i)(θ)=−logP(y(i)∣x(i);θ)=−log∑s=1k​exp(hˉθ​(x(i))s​)exp(hˉθ​(x(i))y(i)​)​

The overall loss is again the average over all training samples.

6. Discriminative vs. Generative Learning Algorithms

Logistic regression is classified as a discriminative algorithm because it models p(y∣x) directly, learning the boundary between classes without modeling the data distribution p(x). This contrasts with generative algorithms that model p(x∣y) and p(y) to classify.

7. Hypothesis Class and Decision Boundaries

The set of all classifiers corresponding to logistic regression forms the hypothesis class H:

H={hθ​:hθ​(x)=1{θTx≥0}}

Here, 1{⋅} denotes the indicator function (output is 1 if condition holds, 0 otherwise). The decision boundary is the hyperplane θTx=0, which is linear in the input space.

8. Learning Algorithm

In practice, logistic regression parameters are learned by maximizing the likelihood or equivalently minimizing the cross-entropy loss using optimization algorithms such as batch gradient descent, stochastic gradient descent, or more advanced variants. The gradient of the loss with respect to θ can be computed explicitly, enabling efficient learning.

9. Extensions and Relations to Other Learning Models

  • Logistic regression can be derived as a Generalized Linear Model (GLM) where the link function is the logit (the inverse of the sigmoid).
  • It is closely related to the perceptron algorithm and linear classifiers, but logistic regression outputs probabilities and has a probabilistic interpretation unlike the perceptron.
  • Logistic regression models can be generalized further as parts of neural network architectures representing hypothesis classes of more complex models.
Categories:

Comments

Post a Comment

Popular posts from this blog

Relation of Model Complexity to Dataset Size

Dr. Rishabh Pathak
Core Concept The relationship between model complexity and dataset size is fundamental in supervised learning, affecting how well a model can learn and generalize. Model complexity refers to the capacity or flexibility of the model to fit a wide variety of functions. Dataset size refers to the number and diversity of training samples available for learning. Key Points 1. Larger Datasets Allow for More Complex Models When your dataset contains more varied data points , you can afford to use more complex models without overfitting. More data points mean more information and variety, enabling the model to learn detailed patterns without fitting noise. Quote from the book: "Relation of Model Complexity to Dataset Size. It’s important to note that model complexity is intimately tied to the variation of inputs contained in your training dataset: the larger variety of data points your dataset contains, the more complex a model you can use without overfitting....
Post a Comment
Read more

Linear Models

Dr. Rishabh Pathak
1. What are Linear Models? Linear models are a class of models that make predictions using a linear function of the input features. The prediction is computed as a weighted sum of the input features plus a bias term. They have been extensively studied over more than a century and remain widely used due to their simplicity, interpretability, and effectiveness in many scenarios. 2. Mathematical Formulation For regression , the general form of a linear model's prediction is: y^ ​ = w0 ​ x0 ​ + w1 ​ x1 ​ + … + wp ​ xp ​ + b where; y^ ​ is the predicted output, xi ​ is the i-th input feature, wi ​ is the learned weight coefficient for feature xi ​ , b is the intercept (bias term), p is the number of features. In vector form: y^ ​ = wTx + b where w = ( w0 ​ , w1 ​ , ... , wp ​ ) and x = ( x0 ​ , x1 ​ , ... , xp ​ ) . 3. Interpretation and Intuition The prediction is a linear combination of features — each feature contributes prop...
Post a Comment
Read more

Predicting Probabilities

Dr. Rishabh Pathak
1. What is Predicting Probabilities? The predict_proba method estimates the probability that a given input belongs to each class. It returns values in the range [0, 1] , representing the model's confidence as probabilities. The sum of predicted probabilities across all classes for a sample is always 1 (i.e., they form a valid probability distribution). 2. Output Shape of predict_proba For binary classification , the shape of the output is (n_samples, 2) : Column 0: Probability of the sample belonging to the negative class. Column 1: Probability of the sample belonging to the positive class. For multiclass classification , the shape is (n_samples, n_classes) , with each column corresponding to the probability of the sample belonging to that class. 3. Interpretation of predict_proba Output The probability reflects how confidently the model believes a data point belongs to each class. For example, in ...
Post a Comment
Read more

Uncertainty Estimates from Classifiers

Dr. Rishabh Pathak
1. Overview of Uncertainty Estimates Many classifiers do more than just output a predicted class label; they also provide a measure of confidence or uncertainty in their predictions. These uncertainty estimates help understand how sure the model is about its decision , which is crucial in real-world applications where different types of errors have different consequences (e.g., medical diagnosis). 2. Why Uncertainty Matters Predictions are often thresholded to produce class labels, but this process discards the underlying probability or decision value. Knowing how confident a classifier is can: Improve decision-making by allowing deferral in uncertain cases. Aid in calibrating models. Help in evaluating the risk associated with predictions. Example: In medical testing, a false negative (missing a disease) can be worse than a false positive (extra test). 3. Methods to Obtain Uncertainty from Classifiers 3.1 ...
Post a Comment
Read more

Uncertainty in Multiclass Classification

Dr. Rishabh Pathak
1. What is Uncertainty in Classification? Uncertainty refers to the model’s confidence or doubt in its predictions. Quantifying uncertainty is important to understand how reliable each prediction is. In multiclass classification , uncertainty estimates provide probabilities over multiple classes, reflecting how sure the model is about each possible class. 2. Methods to Estimate Uncertainty in Multiclass Classification Most multiclass classifiers provide methods such as: predict_proba: Returns a probability distribution across all classes. decision_function: Returns scores or margins for each class (sometimes called raw or uncalibrated confidence scores). The probability distribution from predict_proba captures the uncertainty by assigning a probability to each class. 3. Shape and Interpretation of predict_proba in Multiclass Output shape: (n_samples, n_classes) Each row corresponds to the probabilities of ...
Post a Comment
Read more