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Vector Norms and Distances

Source: R/vector_functionals.R
vector_functionals.Rd

Functions to compute various vector norms, distances, and functional properties. These functionals form the basis of many numerical analysis and signal processing algorithms.

Norms (Magnitude):

  • l1_norm: Computes the L1 norm (Manhattan norm), sum of absolute values.

  • l2_norm: Computes the L2 norm (Euclidean norm), square root of sum of squares.

  • sup_norm: Computes the Supremum (L-infinity) norm, the maximum absolute value.

  • lp_norm: Computes the Lp norm for an arbitrary integer p.

Distances (Difference):

  • l1_distance: Computes the L1 distance between two vectors.

  • l2_distance: Computes the L2 (Euclidean) distance between two vectors.

  • sup_distance: Computes the Supremum (L-infinity) distance/Chebyshev distance.

  • lp_distance: Computes the Lp distance for an arbitrary integer p.

Sparsity & Structure:

  • l0_pseudo_norm: Counts the number of non-zero elements (Hamming weight).

  • hamming_distance: Counts the number of mismatching elements between two vectors.

  • total_variation: Computes the total variation (sum of absolute differences between adjacent elements).

Usage

l0_pseudo_norm(x)

hamming_distance(x, y)

l1_norm(x)

l1_distance(x, y)

l2_norm(x)

l2_distance(x, y)

sup_norm(x)

sup_distance(x, y)

lp_norm(x, p)

lp_distance(x, y, p)

total_variation(x)

Arguments

x

A numeric vector.

y

A numeric vector of the same length as x (for distance functions).

p

A positive integer indicating the order of the norm or distance (for Lp functions).

Value

A single numeric value with the computed norm or distance.

Details

  • L0 Pseudo-Norm: Not a true norm (doesn't satisfy homogeneity), but useful for sparsity (e.g., Compressed Sensing).

  • L1 Norm: Often used in sparse signal recovery (LASSO).

  • Total Variation: Useful in signal processing for denoising while preserving edges (Total Variation Denoising).

The implementations are designed to be efficient and work with various numeric types.

References

Higham, N. J. (2002). Accuracy and stability of numerical algorithms. SIAM. Mallat, S. (2008). A wavelet tour of signal processing: the sparse way. Academic press.

See also

Boost Documentation

Examples

v1 <- c(1, -2, 3)
v2 <- c(4, -5, 6)

# --- Norms ---
l1_norm(v1)      # |1| + |-2| + |3| = 6
#> [1] 6
l2_norm(v1)      # sqrt(1^2 + (-2)^2 + 3^2) = sqrt(14)
#> [1] 3.741657
sup_norm(v1)     # max(|1|, |-2|, |3|) = 3
#> [1] 3
lp_norm(v1, 3)   # Cube root of sum of cubes
#> [1] 3.301927

# --- Distances ---
l1_distance(v1, v2)
#> [1] 9
l2_distance(v1, v2)
#> [1] 5.196152
hamming_distance(c(1, 0, 1), c(0, 1, 1)) # 2 differences (pos 1 and 2)
#> [1] 2

# --- Structure ---
l0_pseudo_norm(c(0, 5, 0, 2)) # Returns 2 (two non-zeros)
#> [1] 2
total_variation(c(1, 5, 2))   # |5-1| + |2-5| = 4 + 3 = 7
#> [1] 7