lnSNln21

lnSNln21 is a theoretical construct in the study of hierarchical, logarithmically scaled networks. It refers to a class of maps formed by alternating logarithmic transformations with a normalization step, with the numeral 21 signaling the intended depth of the recursive construction. The name encodes its signature operations: ln for logarithmic scaling, SN for sign-normalization, and a final ln for compression.

Formally, lnSNln21 denotes the repeated application of a map f on a real-valued vector x in R^d, defined by f(x) = L(SN(L(x))), iterated to a depth of 21 layers. L denotes a componentwise logarithmic transformation applied after rescaling inputs to a positive range; SN is a sign-normalization that stabilizes sign variation; the concluding L reintroduces a log-scale to compress the range. The 21 specifies the number of outer iterations. The construction is designed to produce a layered representation that blends compression from the log with stabilization from normalization, yielding a hierarchical structure.

Properties of the construction include stabilization of variance across layers and a tendency toward bounded outputs under wide input ranges. Computationally, each layer scales linearly with the dimension, offering predictable cost for moderate datasets. The interplay between logarithmic compression and normalization drives its characteristic behavior, supporting analyses of stability and invertibility in multi-layer transforms.

lnSNln21 is primarily of theoretical or pedagogical interest and is used to illustrate how multiple scale transformations interact in hierarchical representations. It appears in discussions about data normalization, fractal-like networks, and the stability of multi-layer transforms. See also logarithm, normalization, hierarchical models.