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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...
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Classification and Logistic Regression

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Dr. Rishabh Pathak

1. Classification Problem

  • Definition: Classification is a supervised learning task where the output variable y is discrete-valued rather than continuous.
  • In particular, consider binary classification where y∈ {0,1} (e.g., spam detection: spam =1, not spam =0).
  • Each training example is a pair (x(i), y(i)), where x(i)∈Rd is a feature vector, and y(i) is the label.

2. Why Not Use Linear Regression for Classification?

  • Linear regression tries to predict continuous values, which is problematic for classification as the prediction can be outside [0,1].
  • For example, predicting y≈1.5 or −0.2 is meaningless when y is binary.
  • Instead, we want the output hθ​(x) to be interpreted as the probability that y=1 given x.

3. Logistic Regression Model

Hypothesis:

hθ​(x)=g(θTx)=1+e−θTx1​,

where:

  • g(z)=1+e−z1​ is the sigmoid function, which maps any real value to the interval (0, 1).
  • θ∈Rd+1 are parameters (including intercept term).
  • hθ​(x) can be interpreted as the estimated probability P(y=1∣x;θ).

Decision Boundary:

  • Predict y=1 if hθ​(x)≥0.5; otherwise, predict y=0.
  • The decision boundary corresponds to θTx=0, which is a linear boundary in input space.

4. Loss Function and Cost Function

Probability Model:

  • Logistic regression models conditional probability directly:

P(y=1∣x;θ)=hθ​(x),P(y=0∣x;θ)=1−hθ​(x).

  • Equivalently, likelihood for data point (x(i),y(i)):

p(y(i)∣x(i);θ)=(hθ​(x(i)))y(i)(1−hθ​(x(i)))1−y(i).

Cost (Loss) Function:

  • Use negative log-likelihood (cross-entropy loss) as cost per example:

J(i)(θ)=−[y(i)loghθ​(x(i))+(1−y(i))log(1−hθ​(x(i)))].

  • Overall cost function (average over n examples):

J(θ)=n1​∑i=1n​J(i)(θ).

  • This loss is convex in θ, enabling efficient optimization.

5. Training Logistic Regression

·         Use methods such as gradient descent or more advanced optimization (Newton's method, quasi-Newton) to minimize cost J(θ).

·         The gradient of the cost function is:

∇θ​J(θ)=n1​∑i=1n​(hθ​(x(i))−y(i))x(i).

  • Update rule in gradient descent:

θ:=θ−α∇θ​J(θ),

where α is the learning rate.

6. Multi-class Classification

·         When y∈{1,2,...,k} for k>2, logistic regression generalizes to multinomial logistic regression or Softmax regression.

·         Model outputs hˉθ​(x)∈Rk called logits.

·         The Softmax function converts logits into probabilities:

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

  • Loss for example (x(i),y(i)) is the negative log-likelihood:

J(i)(θ)=−logP(y(i)∣x(i);θ).

7. Discriminative vs. Generative Classification Algorithms

  • Discriminative algorithms (like logistic regression) model p(y∣x) directly or learn a direct mapping from x to y.
  • Generative algorithms model the joint distribution p(x,y)=p(x∣y)p(y).
  • Example: Gaussian Discriminant Analysis (GDA).
  • Logistic regression is an example of a discriminative approach focusing purely on p(y∣x).

8. Linear Hypothesis Class and Decision Boundaries

  • Logistic regression hypothesis class:

H={hθ​:hθ​(x)=1{θTx≥0}​},

which are classifiers with linear decision boundaries.

  • More generally, hypothesis classes can be extended to neural networks or other complex architectures.

9. Perceptron Learning as Contrast to Logistic Regression

·         Perceptron also uses a linear classifier but with a different loss and update rule.

·         Logistic regression provides probabilistic outputs and optimizes a convex cost function, generally yielding better statistical properties.

10. Practical Considerations

  • Feature scaling often improves numerical stability.
  • Regularization (e.g., L2) is frequently added to cost to prevent overfitting.
  • Logistic regression handles input features linearly; non-linear boundaries require feature engineering or kernel methods.

Summary:

Logistic regression is a fundamental classification algorithm that models the conditional probability of the positive class using a sigmoid of a linear function of input features. It is trained via maximizing likelihood (or minimizing cross-entropy loss) and extends naturally to multi-class problems via Softmax. It is a discriminative model focusing directly on p(y∣x) and yields linear decision boundaries. It contrasts with generative models by its direct approach to classification.

 

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