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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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Unrestricted Sampling

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Dr. Rishabh Pathak

Unrestricted sampling, also known as simple random sampling, is a fundamental sampling technique where each element in the population has an equal and independent chance of being selected for the sample. In unrestricted sampling:


1.    Equal Probability of Selection:

§  In simple random sampling, every element in the population has an equal probability of being chosen for the sample. This ensures that each unit is selected independently of other units, without any bias towards specific elements.

2.    Random Selection:

§  The selection of sample elements is done randomly, without any systematic pattern or predetermined order. This randomness is essential to ensure that the sample is representative of the population and to minimize selection bias.

3.    Independence of Selection:

§  Each selection is made independently of previous selections, meaning that the inclusion or exclusion of one element does not influence the selection of other elements. This independence helps maintain the randomness of the sample.

4.    Statistical Validity:

§  Simple random sampling is a statistically valid method that allows researchers to make inferences about the population based on the characteristics of the sample. It provides a basis for estimating population parameters and assessing the precision of the results.

5.    Efficiency and Simplicity:

§  Unrestricted sampling is straightforward to implement and analyze, making it an efficient sampling method for many research studies. It does not require complex stratification or clustering procedures, which can simplify the sampling process.

6.    Representativeness:

§  When conducted properly, simple random sampling can produce a sample that is representative of the population, allowing researchers to generalize their findings with confidence. This representativeness is crucial for drawing valid conclusions from the sample data.

7.    Sampling Error:

§  Despite its advantages, simple random sampling may still be subject to sampling error, which is the variability between sample estimates and population parameters. Researchers should account for sampling error when interpreting the results of a simple random sample.

Overall, unrestricted sampling through simple random sampling is a foundational and widely used technique in research methodology. By ensuring randomness and equal probability of selection, researchers can create samples that are unbiased, representative, and suitable for making valid inferences about the population.

 

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