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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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Purposive Sampling

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Dr. Rishabh Pathak

Purposive sampling, also known as judgmental or selective sampling, is a non-probability sampling technique where sample units are selected based on specific criteria determined by the researcher's judgment and purpose of the study. This method involves the deliberate selection of sample units that possess certain characteristics of interest. Here are some key points about purposive sampling:


1.    Definition:

o    Purposive sampling is a sampling method where researchers select sample units based on specific criteria or characteristics relevant to the research objectives.

o    Sample units are chosen intentionally to represent certain traits, experiences, or variations within the population.

2.    Characteristics:

o    Purposive sampling is a non-random sampling technique that relies on the researcher's expertise, judgment, and knowledge of the population.

o    Researchers use their discretion to select sample units that are most likely to provide valuable insights or represent the diversity of the population.

3.    Types of Purposive Sampling:

o Maximum Variation Sampling: Selecting sample units that represent a wide range of characteristics or experiences within the population.

o Homogeneous Sampling: Choosing sample units that share common characteristics or traits to study a specific subgroup.

o    Expert Sampling: Selecting sample units based on the expertise or knowledge they possess related to the research topic.

o Typical Case Sampling: Choosing sample units that are considered typical or representative of the population.

4.    Advantages:

o  Purposive sampling allows researchers to focus on specific characteristics or subgroups of interest, making it suitable for targeted research objectives.

o    This method is valuable for studying rare populations, exploring specific phenomena, or gaining in-depth insights into particular traits.

5.    Limitations:

o    Results obtained from purposive samples may not be generalizable to the entire population due to selection bias and non-random selection.

o The subjective nature of purposive sampling can introduce researcher bias and limit the external validity of the findings.

6.    Applications:

o    Purposive sampling is commonly used in qualitative research, case studies, ethnographic studies, and situations where specific characteristics or experiences are of interest.

o This method is particularly useful when studying unique populations, exploring diverse perspectives, or conducting in-depth investigations.

7.    Considerations:

o    Researchers should clearly define the criteria for selecting sample units in purposive sampling and justify their choices based on the research objectives.

o    While purposive sampling offers flexibility and targeted sampling, researchers should acknowledge its limitations in terms of generalizability and potential bias.

Purposive sampling is a valuable sampling technique that allows researchers to strategically select sample units based on specific criteria relevant to their research goals. While this method offers advantages in terms of targeted sampling and in-depth exploration, researchers should be mindful of its limitations in terms of representativeness and potential bias. Careful consideration of the research objectives and criteria for sample selection is essential when employing purposive sampling in a study.

 

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