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Energy Manifold Natural Gradient Descent: From Riemannian Optimization to Modern Neuroscience, NeuroAI and Quantum Physics

Dr. Rishabh Pathak
When Geometry, Energy, Artificial Intelligence and Neuroscience Converge Modern Artificial Intelligence is rapidly moving beyond the idea that learning simply means minimizing an error function. Increasingly, researchers are asking a deeper question: what is the structure of the space in which learning takes place? This question becomes particularly important when the system being modelled is constrained, nonlinear, dynamic, or governed by physical principles. A recent work titled “Energy Manifold Natural Gradient Descent: Riemannian Optimization for Neural PDE Solvers” , by Zhangyong Liang and Huanhuan Gao, introduces Energy Manifold Natural Gradient Descent (EMNGD) , a mathematical framework that extends energy-based natural-gradient optimization from unconstrained Euclidean parameter spaces to constrained Riemannian parameter manifolds . At its core, the framework proposes a simple but powerful principle: An optimization algorithm should not only determine how to reduce error; it sh...
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Simple Random Sampling Without Replacement

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Dr. Rishabh Pathak

Simple random sampling without replacement is a fundamental sampling technique used in research to select a subset of items from a larger population in such a way that each item has an equal probability of being chosen, and once an item is selected, it is not replaced back into the population. Here is an overview of how simple random sampling without replacement works:


1.    Population and Sampling Frame:

§  The population refers to the entire group of interest from which the sample will be drawn. A sampling frame is a list or representation of all the elements in the population that are accessible for sampling.

2.    Assigning Numbers:

§  Each element in the population is assigned a unique identifier or number. These numbers are used to distinguish and select individual items during the sampling process.

3.    Random Selection:

§  To conduct simple random sampling without replacement, researchers use a random selection method to choose items from the population. This can be done using random number tables, software, or other randomization techniques.

4.    Selection Process:

§  Researchers start by selecting a random starting point in the sampling frame. They then proceed to select items systematically based on a random pattern, ensuring that each item has an equal chance of being chosen.

5.    Sample Size:

§  The sample size is predetermined based on the research objectives and statistical considerations. In simple random sampling without replacement, each selected item reduces the pool of available items for subsequent selections.

6.    Representativeness:

§  By ensuring that each item in the population has an equal probability of being included in the sample, simple random sampling without replacement helps in creating a representative sample that reflects the characteristics of the larger population.

7.    Statistical Analysis:

§  Once the sample is selected, researchers can analyze the sample data using various statistical methods to draw conclusions and make inferences about the population. The results obtained from the sample can be generalized to the population with appropriate statistical techniques.

8.    Advantages:

§  Simple random sampling without replacement is straightforward, easy to understand, and helps in reducing bias in the sample selection process. It provides a basis for statistical inference and allows researchers to estimate population parameters with known precision.

9.    Limitations:

§  One limitation of simple random sampling without replacement is that it may not be practical for very large populations, as the process of selecting samples without replacement can become cumbersome. In such cases, other sampling methods like stratified sampling or cluster sampling may be more efficient.

Simple random sampling without replacement is a foundational sampling method that forms the basis for many other sampling techniques. By following the principles of randomness and equal probability, researchers can ensure the validity and reliability of their research findings when using this sampling approach.

 

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