Update docs.

Author: kernelmethod

docs/make.jl

@@ -21,7 +21,7 @@ makedocs(
                 "Function-space hashing" => "function_hashing.md",
                 "Performance tips" => "performance.md",
                 "FAQ" => "faq.md",
-                "Notation and glossary" => "notation_and_glossary.md",
+                "Glossary" => "glossary.md",
                 "API reference" => "full_api.md",
                ]
 )

docs/src/function_hashing.md

@@ -19,7 +19,8 @@ julia> hashfn(x -> 5x^3 - 2x^2 - 9x + 1)
  1
 ```
 
-## Similarity statistics in ``L_{\mu}^p`` function spaces
+## Similarity statistics in function spaces
+The LSHFunctions module currently supports the following similarity statistics for function spaces. Unless otherwise stated, all functions are assumed to be members of an [``L^p_{\mu}(\Omega)`` function space](https://en.wikipedia.org/wiki/Lp_space).
 
 ### ``L_{\mu}^p`` distance
 
@@ -33,13 +34,7 @@ julia> hashfn(x -> 5x^3 - 2x^2 - 9x + 1)
 \left\langle f, g\right\rangle_{L_{\mu}^2} = \int_{\Omega} f(x)g(x) \hspace{0.15cm} d\mu(x)
 ```
 
-When `f(x)` and `g(x)` are allowed to take on complex values, the inner product is defined as
-
-```math
-\left\langle f, g\right\rangle_{L_{\mu}^2} = \int_{\Omega} f(x)\overline{g(x)} \hspace{0.15cm} d\mu(x)
-```
-
-where ``\overline{g(x)}`` is the complex conjugate of ``g(x)``.
+When ``f`` and ``g`` are allowed to take on complex values, ``g(x)`` is replaced by ``\overline{g(x)}`` (the complex conjugate of ``g(x)``) in the formula above.
 
 ### Cosine similarity
 ```math

docs/src/glossary.md

@@ -0,0 +1,37 @@
+# LSHFunctions notation and glossary
+
+!!! warning "Under construction"
+    This section is currently being developed. If you're interested in helping write this section, feel free to [open a pull request](https://github.com/kernelmethod/LSHFunctions.jl/pulls); otherwise, please check back later.
+
+## Terms
+
+---
+
+**LSH**: an acronym for locality-sensitive hashing.
+
+---
+
+``L^p_{\mu}(\Omega)`` **function space** ([wikipedia](https://en.wikipedia.org/wiki/Lp_space)): a set of functions[^1] whose inputs come from some set ``\Omega`` and whose outputs are either real or complex numbers. ``\mu`` is a [measure](https://en.wikipedia.org/wiki/Measure_space) and ``p`` is a positive number. The ``L^p_{\mu}(\Omega)`` [norm](https://en.wikipedia.org/wiki/Norm_(mathematics)), denoted with ``\|\cdot\|_{L^p_{\mu}}`` (where ``\Omega`` is implicit), is defined as
+
+```math
+\|f\|_{L^p_{\mu}} = \left(\int_{\Omega} \left|f(x)\right|^p \hspace{0.15cm} d\mu(x)\right)^{1/p}
+```
+
+In the case where ``p = 2``, there is also an [inner product](https://en.wikipedia.org/wiki/Inner_product_space) defined for the space:
+
+```math
+\left\langle f, g\right\rangle = \int_{\Omega} f(x)\overline{g(x)} \hspace{0.15cm} d\mu(x)
+```
+
+where ``\overline{g(x)}`` is the complex conjugate of ``g(x)``. A function in ``L^p_{\mu}(\Omega)`` must have the property that ``\|f\|_{L^p_{\mu}}`` is finite.
+
+*Example*: ``f(x) = x^2 - 3x + 2`` is a function in ``L^2([-1,1])`` (with ``\mu`` chosen to be [Lebesgue measure](https://en.wikipedia.org/wiki/Lebesgue_measure)) because ``\|f\|_{L^2} = \sqrt{\int_{-1}^1 \left|f(x)\right|^2 \hspace{0.15cm} dx}`` is finite. However, it is *not* a function in ``L^2([-\infty,\infty])`` because the ``\|f\|_{L^2} = \sqrt{\int_{-\infty}^{\infty} \left|f(x)\right|^2 \hspace{0.15cm} dx`` is infinite.
+
+---
+
+**Similarity statistic**: a number that represents the similarity between two data points. Different similarity statistics have different ways of defining what "similar" means.
+
+A similarity statistic can be interpreted in many different ways; for instance, cosine similarity is defined between -1 and 1, with *higher* values indicating *higher* similarity. Meanwhile, ``\ell^p`` distance is defined between 0 and ``+\infty``, with *higher* distances indicating *lower* similarity.
+
+## Footnotes
+[^1]: technically, equivalence classes of functions.

docs/src/notation_and_glossary.md

@@ -29,7 +29,7 @@ true
 ```
 
 ## SimHash
-*SimHash*[^1][^2] is a family of LSH functions for hashing with respect to cosine similarity. You can generate a new hash function from this family by calling [`SimHash`](@ref):
+*SimHash*[^1][^Charikar02] is a family of LSH functions for hashing with respect to cosine similarity. You can generate a new hash function from this family by calling [`SimHash`](@ref):
 
 ```jldoctest; setup = :(using LSHFunctions)
 julia> hashfn = SimHash();
@@ -123,6 +123,6 @@ savefig("simhash_collision_probability.svg")
 
 ### Footnotes
 
-[^1]: Moses S. Charikar. *Similarity estimation techniques from rounding algorithms*. In Proceedings of the Thiry-Fourth Annual ACM Symposium on Theory of Computing, STOC '02, page 380–388, New York, NY, USA, 2002. Association for Computing Machinery. 10.1145/509907.509965.
+[^1]: [`SimHash` API reference](@ref SimHash)
 
-[^2]: [`SimHash` API reference](@ref SimHash)
+[^Charikar02]: Moses S. Charikar. *Similarity estimation techniques from rounding algorithms*. In Proceedings of the Thiry-Fourth Annual ACM Symposium on Theory of Computing, STOC '02, page 380–388, New York, NY, USA, 2002. Association for Computing Machinery. 10.1145/509907.509965.