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Linear Approximation in Frequency Domain

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Linear systems are characterized by two main properties: superposition and homogeneity. Superposition allows the response to multiple inputs to be the sum of the responses to each individual input. Homogeneity ensures that scaling an input by a scalar results in the response being scaled by the same scalar.
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It is far more common for collisions to occur in two dimensions; that is, the initial velocity vectors are neither parallel nor antiparallel to each other. Let's see what complications arise from this. The first idea is that momentum is a vector. Like all vectors, it can be expressed as a sum of perpendicular components (usually, though not always, an x-component and a y-component, and a z-component if necessary). Thus, when the statement of conservation of momentum is written for a...
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The dot product is a powerful tool in problem-solving involving vectors, given that the dot product of two vectors is the product of their magnitudes and the cosine of the angle between them measured anti-clockwise. Solving problems involving the dot product requires understanding its properties and developing a step-by-step process to solve them. Here are the main steps to follow when solving any general problem involving the dot product:
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In mechanical engineering, one-degree-of-freedom systems form the basis of a wide range of electrical and mechanical components. Using these models, engineers can predict the behavior of various parts in a larger system, which gives them insight into how different forces interact with each other.
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Updated: May 24, 2025

Augmenting Large Language Models via Vector Embeddings to Improve Domain-Specific Responsiveness
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Take Your Pick: Enabling Effective Distributed Learning Within Low-Dimensional Feature Space.

Guogang Zhu, Xuefeng Liu, Shaojie Tang

    IEEE Transactions on Neural Networks and Learning Systems
    |March 3, 2025
    PubMed
    Summary
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    FedPick enhances personalized federated learning (PFL) by adaptively selecting task-relevant features in the low-dimensional feature space. This approach improves cross-domain model performance and interpretability compared to parameter-space methods.

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    Area of Science:

    • Artificial Intelligence
    • Machine Learning
    • Distributed Systems

    Background:

    • Personalized federated learning (PFL) enables diverse client models for cross-domain applications like autonomous driving and medical diagnosis.
    • Current PFL models use a global encoder for universal features and personalized layers, but domain gaps cause irrelevant feature components.
    • Existing methods personalizing encoder parameters face challenges due to high dimensionality and nonlinearity.

    Purpose of the Study:

    • To propose FedPick, a novel PFL framework operating in the low-dimensional feature space.
    • To address the challenge of irrelevant universal features in cross-domain PFL by adaptively selecting task-relevant features.
    • To provide a more accessible and interpretable PFL implementation.

    Main Methods:

    • FedPick adaptively selects task-relevant features from global encoder outputs based on local data distribution.
    • The framework operates within the lower-dimensional feature space, offering greater intuitiveness and interpretability.
    • Feature selection is performed client-wise to tailor universal features to local tasks.

    Main Results:

    • FedPick effectively selects task-relevant features for each client in cross-domain scenarios.
    • Experimental results demonstrate significant improvements in model performance across multiple datasets.
    • The proposed method shows superior performance compared to existing PFL techniques.

    Conclusions:

    • FedPick offers an effective and interpretable solution for personalized federated learning in cross-domain settings.
    • Adaptive feature selection in the feature space is a viable alternative to parameter personalization.
    • The framework has strong potential for applications requiring robust cross-domain learning.