Hyper-Indexing with LSHash (Locality Sensitive Hashing)

Indexing topic vectors from an LSI Model is more difficult than it seems. My first instinct was to use the 3D indexer plugin for PostgreSQL, PostGIS. After all that’s the typical example I keep in my head for indexing. You create a discrete “on or off” label for each location based on whether it is present or absent within a grid point. This allows you to efficiently find it (and any nearby points) with a query with a WHERE grid = 'A11' for a letter/int 2D indexing system that you see on old paper road maps from AAA.

I assumed this simple approach would extend out to multiple dimensions, but it doesn’t. Turns out, it’s not possible to efficiently index and search on hyperdimensional space (out beyond a few dimensions). But you can push the limits with a python package calls LSHash which implements Locality Sensitive Hashing. This approach isn’t theoretically as efficent as R*Tree (the current state of the art for multi-dimensional indexing), but it’s pretty darn powerful up to 8 dimensions.

import numpy as np
from lshash import LSHash

tenthclosest = []  # for each test, record distance to the 10th closest point for a random query
for D in range(2, 11):  # Run tests for 2D through 10D
    X = np.random.normal(size=(200000, D))  # Fill the N-D space with 200k random vectors
    lsh = LSHash(hash_size=int(64 / D) + D, input_dim=D, num_hashtables=D)

    # query vector
    q = np.random.normal(size=(D,))
    q /= np.linalg.norm(q)

    distances = []
    for x in X:
        x /= np.linalg.norm(x)  # topic vectors are typically normalized
        distances += [1 - np.sum(x * q)]  # keep track of all the cosine distances to double check

    distances = sorted(distances)
    closest = lsh.query(x, distance_func='cosine')
    N = len(closest)
    rank = min(10, N)
    tenthclosest += [[D, N - 1, closest[rank - 1][-1] if N else None, distances[rank - 1]]]

For 8D space and higher you’ll need millions of points in your space to have a chance of finding anything nearby. And the lshash will be less and less useful at these higher dimensions because it won’t be possible to partition the space with a reasonable number of hyperplanes so that the returned points truly are the closest points.

>>> tenthclosest
  [2, 9480, 1.8566308490619576e-09, 1.8566308490619576e-09]
  [3, 1791, 9.3939812061627492e-05, 9.3939812061738515e-05]
  [4, 2492, 0.0016495388314403669, 0.0016495388314402559]
  [5, 1042, 0.0062851055257979738, 0.0062851055257979738]
  [6, 1161, 0.01919247547800762, 0.01919247547800762]
  [7, 1298, 0.037779915230408245, 0.037190703103046285]   # the hashes didn't find them all
  [8, 932, 0.040481502423105997, 0.040481502423105997]
  [9, 1230, 0.066732165658179299, 0.066732165658179188]   # the hashes didn't find them all
  [10, 835, 0.10840345965900211, 0.095584594509859677]    # the hashes didn't find them all

So the lesson here is.

  • LSH partitions the space with hyperplanes through the origin, so it works best for normalized vectors
  • As the number of dimensions increases your need more hashtables to ensure you can find points in neighboring grid cells in case some queries lie near the boundary of a grid in one of the hashtables
  • As the number of dimensions increases the number of hyperplanes you need to slice the space decreases proportionately
  • As the number of dimensions incrases the distance to the nearest points will increase exponentially, even for dimension-insensitive metrics like cosine distance.
Written on November 12, 2016