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Implicit Differentiation: Problem Solving01:29

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Curves defined implicitly, where variables cannot be separated algebraically, require specialized techniques for analysis. The conchoid of Nicomedes exemplifies such a case. Its equation links x and y in a way that prevents isolation of one variable, making implicit differentiation essential to determine the slope and behavior at any point on the curve.The implicit form of the conchoid can be expressed as:To differentiate this equation, y is treated as a function of x, and the chain rule is...
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Related Experiment Video

Updated: Feb 25, 2026

Deep Neural Networks for Image-Based Dietary Assessment
13:19

Deep Neural Networks for Image-Based Dietary Assessment

Published on: March 13, 2021

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Privacy Preservation in Distributed Subgradient Optimization Algorithms.

Youcheng Lou, Lean Yu, Shouyang Wang

    IEEE Transactions on Cybernetics
    |August 3, 2017
    PubMed
    Summary
    This summary is machine-generated.

    This study introduces a novel privacy-preserving distributed subgradient optimization algorithm. It protects sensitive data by using asynchronous updates and projection mechanisms, unlike older methods.

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    Last Updated: Feb 25, 2026

    Deep Neural Networks for Image-Based Dietary Assessment
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    Deep Neural Networks for Image-Based Dietary Assessment

    Published on: March 13, 2021

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    Area of Science:

    • Computer Science
    • Optimization Theory
    • Cybersecurity

    Background:

    • Distributed optimization algorithms often prioritize performance over privacy.
    • Existing algorithms may expose sensitive agent data (subgradients) to malicious participants.
    • Privacy is crucial for applications handling confidential information.

    Purpose of the Study:

    • To analyze the privacy vulnerabilities of synchronous distributed subgradient algorithms.
    • To propose a new asynchronous algorithm that enhances agent privacy.
    • To establish the convergence and optimality of the proposed privacy-preserving method.

    Main Methods:

    • Demonstrated privacy leakage in synchronous homogeneous-stepsize algorithms.
    • Developed a distributed subgradient asynchronous heterogeneous-stepsize projection algorithm.
    • Analyzed convergence and optimality properties of the new algorithm.

    Main Results:

    • The synchronous algorithm allows malicious agents to infer others' subgradients.
    • The proposed asynchronous algorithm effectively protects agent privacy.
    • Convergence and optimality are proven for the new privacy-preserving approach.

    Conclusions:

    • Synchronous distributed subgradient algorithms pose privacy risks.
    • Asynchronous heterogeneous-stepsize projection offers robust privacy preservation.
    • The novel algorithm balances distributed optimization efficiency with strong data security.