Jove
Visualize
Contact Us
JoVE
x logofacebook logolinkedin logoyoutube logo
ABOUT JoVE
OverviewLeadershipBlogJoVE Help Center
AUTHORS
Publishing ProcessEditorial BoardScope & PoliciesPeer ReviewFAQSubmit
LIBRARIANS
TestimonialsSubscriptionsAccessResourcesLibrary Advisory BoardFAQ
RESEARCH
JoVE JournalMethods CollectionsJoVE Encyclopedia of ExperimentsArchive
EDUCATION
JoVE CoreJoVE BusinessJoVE Science EducationJoVE Lab ManualFaculty Resource CenterFaculty Site
Terms & Conditions of Use
Privacy Policy
Policies

Related Concept Videos

Buffers: Buffer Capacity01:09

Buffers: Buffer Capacity

Buffer capacity is the quantitative measure of a buffer to resist the change in pH. As shown in the following equation, the buffer capacity, denoted by 'beta', is expressed as the number of moles of acid or base needed to change the pH of a one-liter buffer solution by 1 unit. Here, Ca and Cb indicate the number of moles of acid and base, respectively. Note that dpH represents the change in pH.
In the graph, pH is plotted as a function of the number of moles of base (Cb) added to a weak acid...
Dimensional Analysis03:40

Dimensional Analysis

Dimensional analysis, also known as the factor label method, is a versatile approach for mathematical operations. The main principle behind this approach is: the units of quantities must be subjected to the same mathematical operations as their associated numbers. This method can be applied to computations ranging from simple unit conversions to more complex and multi-step calculations involving several different quantities and their units.
Conversion Factors and Dimensional Analysis
The unit...
Dimensional Analysis01:27

Dimensional Analysis

Dimensional analysis is a valuable technique in fluid mechanics for simplifying complex problems by reducing them into dimensionless groups. These groups capture the essential relationships between the variables involved, allowing researchers and engineers to analyze fluid flow without dealing with each variable individually. This approach reduces the number of independent variables, allowing for easier analysis and better understanding of physical phenomena.
In fluid mechanics, dimensional...
Dimensional Analysis02:19

Dimensional Analysis

The concept of dimension is important because every mathematical equation linking physical quantities must be dimensionally consistent, implying that mathematical equations must meet the following two rules. The first rule is that, in an equation, the expressions on each side of the equal sign must have the same dimensions. This is fairly intuitive since we can only add or subtract quantities of the same type (dimension). The second rule states that, in an equation, the arguments of any of the...
Dimensional Analysis01:23

Dimensional Analysis

Dimensional analysis is a powerful tool that is used in physics and engineering to understand and predict the behavior of physical systems. The basic idea behind dimensional analysis is to express physical quantities in terms of fundamental dimensions such as the mass, length, and time. Derived dimensions like the velocity, acceleration, and force are derived from the combinations of these fundamental dimensions.
Dimensional analysis allows us to analyze and compare physical quantities on a...
Boundary Conditions: Lossless Lines01:21

Boundary Conditions: Lossless Lines

Consider a single-phase, two-wire, lossless transmission line terminated by an impedance at the receiving end and a source with Thevenin voltage and impedance at the sending end. The line, with length, has a surge impedance and wave velocity determined by the line's inductance and capacitance.
At the receiving end, the boundary condition states that the voltage equals the product of the receiving-end impedance and current. This relationship is expressed as a function of the incident and...

You might also read

Related Articles

Articles linked to this work by shared authors, journal, and citation graph.

Sort by
Same author

Bayesian Linear Inverse Problems in Regularity Scales with Discrete Observations.

Sankhya. Series A. (2008)·2024
Same author

Bi-<i>s</i>*-Concave Distributions.

Journal of statistical planning and inference·2021
Same author

Adaptive Bayesian credible bands in regression with a Gaussian process prior.

Sankhya. Series A. (2008)·2020
Same author

HIGHER ORDER ESTIMATING EQUATIONS FOR HIGH-DIMENSIONAL MODELS.

Annals of statistics·2019
Same author

The Bennett-Orlicz Norm.

Sankhya. Series A. (2008)·2018
Same author

Entropy of Convex Functions on ℝ <i></i>.

Constructive approximation·2018
Same journal

High dimensional Bernstein-von Mises: simple examples.

Institute of Mathematical Statistics collections·2010
Same journal

Model selection and sensitivity analysis for sequence pattern models.

Institute of Mathematical Statistics collections·2010
See all related articles

Related Experiment Video

Updated: Jun 10, 2026

Volume Segmentation and Analysis of Biological Materials Using SuRVoS (Super-region Volume Segmentation) Workbench
11:38

Volume Segmentation and Analysis of Biological Materials Using SuRVoS (Super-region Volume Segmentation) Workbench

Published on: August 23, 2017

A note on bounds for VC dimensions.

Aad van der Vaart1, Jon A Wellner

  • 1Department of Mathematics, Faculty of Sciences, Vrije Universiteit De Boelelaan 1081a, 1081 HV Amsterdam, aad@cs.vu.nl.

Institute of Mathematical Statistics Collections
|September 28, 2011
PubMed
Summary
This summary is machine-generated.

This study establishes bounds for the Vapnik-Chervonenkis (VC) dimension of set classes created through unions, intersections, and products of multiple VC classes. These findings are crucial for understanding the complexity of learnable sets in machine learning theory.

More Related Videos

Motion-Acuity Test for Visual Field Acuity Measurement with Motion-Defined Shapes
06:25

Motion-Acuity Test for Visual Field Acuity Measurement with Motion-Defined Shapes

Published on: February 23, 2024

Related Experiment Videos

Last Updated: Jun 10, 2026

Volume Segmentation and Analysis of Biological Materials Using SuRVoS (Super-region Volume Segmentation) Workbench
11:38

Volume Segmentation and Analysis of Biological Materials Using SuRVoS (Super-region Volume Segmentation) Workbench

Published on: August 23, 2017

Motion-Acuity Test for Visual Field Acuity Measurement with Motion-Defined Shapes
06:25

Motion-Acuity Test for Visual Field Acuity Measurement with Motion-Defined Shapes

Published on: February 23, 2024

Area of Science:

  • Machine Learning Theory
  • Computational Learning Theory
  • Set Theory

Background:

  • The Vapnik-Chervonenkis (VC) dimension is a fundamental measure of the capacity of a class of sets.
  • Understanding how operations like union, intersection, and product affect VC dimension is key to analyzing complex hypothesis spaces.
  • Existing bounds are often specific to certain operations or a limited number of set classes.

Purpose of the Study:

  • To derive general bounds for the VC dimension of set classes formed by combining multiple base VC classes.
  • To investigate the impact of set operations (union, intersection, product) on the overall VC dimension.
  • To provide a theoretical framework for analyzing the complexity of function classes generated by these operations.

Main Methods:

  • Utilizing established techniques from VC theory to analyze set operations.
  • Developing combinatorial arguments to bound the growth rate of combined set classes.
  • Applying induction and structural analysis on the set classes.

Main Results:

  • Established upper bounds for the VC dimension of unions, intersections, and products of m VC classes.
  • Demonstrated how the VC dimension scales with the number of base classes (m) and their individual VC dimensions.
  • Provided explicit formulas or inequalities for the resulting VC dimensions.

Conclusions:

  • The VC dimension of combined set classes can be effectively bounded, offering insights into their learnability.
  • The derived bounds are applicable to a wide range of machine learning models whose hypothesis spaces can be represented as combinations of simpler classes.
  • This work contributes to a deeper theoretical understanding of hypothesis space complexity in machine learning.