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

Kernelized Support Vector Machines

Image
Dr. Rishabh Pathak

1. Introduction to SVMs

  • Support Vector Machines (SVMs) are supervised learning algorithms primarily used for classification (and regression with SVR).
  • They aim to find the optimal separating hyperplane that maximizes the margin between classes for linearly separable data.
  • Basic (linear) SVMs operate in the original feature space, producing linear decision boundaries.

2. Limitations of Linear SVMs

  • Linear SVMs have limited flexibility as their decision boundaries are hyperplanes.
  • Many real-world problems require more complex, non-linear decision boundaries that linear SVM cannot provide.

3. Kernel Trick: Overcoming Non-linearity

  • To allow non-linear decision boundaries, SVMs exploit the kernel trick.
  • The kernel trick implicitly maps input data into a higher-dimensional feature space where linear separation might be possible, without explicitly performing the costly mapping.

How the Kernel Trick Works:

  • Instead of computing the coordinates of data points in high-dimensional space (which could be infinite-dimensional), SVM calculates inner products (similarity measures) directly using kernel functions.
  • These inner products correspond to an implicit mapping into the higher-dimensional space.
  • This avoids the curse of dimensionality and reduces computational cost.

4. Types of Kernels

The most common kernels:

1.      Polynomial Kernel

  • Computes all polynomial combinations of features up to a specified degree.
  • Enables capturing interactions and higher-order feature terms.
  • Example: kernel corresponds to sums like feature1², feature1 × feature2⁵, etc..

2.     Radial Basis Function (RBF) Kernel (Gaussian Kernel)

  • Corresponds to an infinite-dimensional feature space.
  • Measures similarity based on the distance between points in original space, decreasing exponentially with distance.
  • Suitable when relationships are highly non-linear and not well captured by polynomial terms.

5. Important Parameters in Kernelized SVMs

1.      Regularization parameter (C)

  • Controls the trade-off between maximizing the margin and minimizing classification error.
  • A small C encourages a wider margin but allows some misclassifications (more regularization).
  • A large C tries to classify all training points correctly but might overfit.

2.     Kernel choice

  • Selecting the appropriate kernel function is critical (polynomial, RBF, linear, etc.).
  • The choice depends on the data and problem structure.

3.     Kernel-specific parameters

  • Each kernel function has parameters:
  • Polynomial kernel: degree of polynomial.
  • RBF kernel: gamma (shape of Gaussian; higher gamma means points closer).
  • These parameters govern the flexibility and complexity of the decision boundary.

6. Strengths and Weaknesses

Strengths

  • Flexibility:
  • SVMs can create complex, non-linear boundaries suitable for both low and high-dimensional data,.
  • Effective in high dimensions:
  • Works well even if the number of features exceeds the number of samples.
  • Kernel trick:
  • Avoids explicit computations in very high-dimensional spaces, saving computational resources.

Weaknesses

  • Scalability:
  • SVMs scale poorly with the number of samples.
  • Practical for datasets up to ~10,000 samples; larger datasets increase runtime and memory significantly.
  • Parameter tuning and preprocessing:
  • Requires careful preprocessing (feature scaling is important), tuning of C, kernel, and kernel-specific parameters for good performance.
  • Interpretability:
  • Model is difficult to interpret; explaining why a prediction was made is challenging.

7. When to Use Kernelized SVMs?

  • Consider kernelized SVMs if:
  • Your features have similar scales or represent homogeneous measurements (e.g., pixel intensities).
  • The dataset is not too large (under ~10,000 samples).
  • You require powerful non-linear classification with well-separated classes.

8. Mathematical Background (Overview)

  • The underlying math is involved and detailed in advanced texts such as The Elements of Statistical Learning by Hastie, Tibshirani, and Friedman.
  • Conceptually:
  • The primal optimization problem tries to maximize the margin while penalizing misclassifications.
  • The dual problem allows the introduction of kernels, enabling use of the kernel trick.

Summary

Aspect

Details

Purpose

Classification with linear or non-linear decision boundaries

Key idea

Map data to higher-dimensional space via kernels (kernel trick)

Common kernels

Polynomial, RBF (Gaussian)

Parameters

Regularization C, kernel type, kernel-specific params (degree, gamma)

Strengths

Flexible decision boundaries, works well in high-dimensions

Weaknesses

Poor scaling to large datasets, requires tuning, less interpretable

Use cases

Data with uniform feature scaling, moderate size datasets

 

Categories:

Comments

Post a Comment

Popular posts from this blog

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

Mesencephalic Locomotor Region (MLR)

Dr. Rishabh Pathak
The Mesencephalic Locomotor Region (MLR) is a region in the midbrain that plays a crucial role in the control of locomotion and rhythmic movements. Here is an overview of the MLR and its significance in neuroscience research and motor control: 1.       Location : o The MLR is located in the mesencephalon, specifically in the midbrain tegmentum, near the aqueduct of Sylvius. o   It encompasses a group of neurons that are involved in coordinating and modulating locomotor activity. 2.      Function : o   Control of Locomotion : The MLR is considered a key center for initiating and regulating locomotor movements, including walking, running, and other rhythmic activities. o Rhythmic Movements : Neurons in the MLR are involved in generating and coordinating rhythmic patterns of muscle activity essential for locomotion. o Integration of Sensory Information : The MLR receives inputs from various sensory modalities and higher brain regions t...
Post a Comment
Read more

Seizures

Dr. Rishabh Pathak
Seizures are episodes of abnormal electrical activity in the brain that can lead to a wide range of symptoms, from subtle changes in awareness to convulsions and loss of consciousness. Understanding seizures and their manifestations is crucial for accurate diagnosis and management. Here is a detailed overview of seizures: 1.       Definition : o A seizure is a transient occurrence of signs and/or symptoms due to abnormal, excessive, or synchronous neuronal activity in the brain. o Seizures can present in various forms, including focal (partial) seizures that originate in a specific area of the brain and generalized seizures that involve both hemispheres of the brain simultaneously. 2.      Classification : o Seizures are classified into different types based on their clinical presentation and EEG findings. Common seizure types include focal seizures, generalized seizures, and seizures of unknown onset. o The classification of seizures is esse...
Post a Comment
Read more

Mu Rhythms compared to Ciganek Rhythms

Dr. Rishabh Pathak
The Mu rhythm and Cigánek rhythm are two distinct EEG patterns with unique characteristics that can be compared based on various features.  1.      Location : o     Mu Rhythm : § The Mu rhythm is maximal at the C3 or C4 electrode, with occasional involvement of the Cz electrode. § It is predominantly observed in the central and precentral regions of the brain. o     Cigánek Rhythm : § The Cigánek rhythm is typically located in the central parasagittal region of the brain. § It is more symmetrically distributed compared to the Mu rhythm. 2.    Frequency : o     Mu Rhythm : §   The Mu rhythm typically exhibits a frequency similar to the alpha rhythm, around 10 Hz. §   Frequencies within the range of 7 to 11 Hz are considered normal for the Mu rhythm. o     Cigánek Rhythm : §   The Cigánek rhythm is slower than the Mu rhythm and is typically outside the alpha frequency range. 3. ...
Post a Comment
Read more

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