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The root mean square of an input vector $$\mathbf{h}$$ with $$d$$ components is mathematically defined as the square root of the mean of its squared elements. It is calculated using the following formula: $$ \sigma_{\mathrm{rms}} = \left(\frac{1}{d} \sum_{k=1}^{d} h_k^2\right)^{\frac{1}{2}} $$

Root Mean Square (RMS) of a Vector

The root mean square (RMS) layer normalization function computes the normalized output by re-scaling the input vector without re-centering it. The mathematical formulation is: $$ \mathrm{LNorm}(\mathbf{h}) = \alpha \cdot \frac{\mathbf{h}}{\sigma_{\mathrm{rms}} + \epsilon} + \beta $$ In this equation, $$\mathbf{h}$$ is the input vector and $$\sigma_{\mathrm{rms}}$$ represents the root mean square of $$\mathbf{h}$$.

RMS Layer Normalization Formula

Consider a d-dimensional vector **h** whose components have a root mean square (RMS) value of σ. A new vector, **h'**, is created by multiplying every component of **h** by a constant factor of 2. What is the RMS of the new vector **h'**?

Without performing the full calculation, analyze the structure of the Root Mean Square (RMS) formula. Based on this analysis, explain which vector (A or B) will have a larger RMS value and why the difference is more significant than a simple comparison of their average component values might suggest.

Impact of an Outlier on Vector Magnitude

A 4-dimensional vector is given by h = [1, 2, 3, 6]. Calculate the root mean square (RMS) of this vector, which is found by taking the square root of the mean of the squares of its components. Round your answer to two decimal places.

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