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Many of LSH families are structured quantizers: they don't take into account underlying statistics

- for example, E2LSH is structured:
- we choose only quantization step $w$ and offset $b$ and have little influence on the density of individual cells
- but can address this issue by learning a Vector Quantizer - such as K-Means: this way we can adapt the cell size to the density of the space in the cell

Unstructured VQ:

- let $\mathcal R \to [ \, 1, 2, \ ... \ , k \, ]$
- and $\mathbf x \to g(\mathbf x) = \mathop{arg min}\limits_{i = 1..k} L_2(\mathbf x, \boldsymbol \mu_i)$
- it maps each vector to a cell indexed by $g(\mathbf x)$
- $k$ is # of possible values of $g(\cdot)$
- $\boldsymbol \mu_i$ are centroids - they define the quantizer
- often learned with K-Means

Illustration:

- Euclidean LSH (Random Projection LSH) vs K-Means
- K-Means adapts to data while others don't
- source: Paulevé2010 figure 3

How to use K-Means to build a LSH?

- generate $L$ different clusterings on the same data by using different seeds
- after this we have $L$ codebooks (sets of centroids) $\{\mathbf c_{j1}, \ ... \ , \mathbf c_{jk}\}$, where $\mathbf c_{ji}$ is $i$th centroid of $j$'s clustering
- each centroid is an $h$ and codebooks is an $g$ in LSH

Indexing:

- a vector to index is assigned to the closest centroid found in a codebook
- use $L$ codebooks to have $L$ cluster assignments

Search time:

- first nearest centroid for each codebook
- then keep only vectors that are assignment to the same centroids

query($\mathbf q$)

- $\text{res} = \varnothing$
- for $j = 1$ to $L$ do
- $i^* = \operatorname{arg\, mix}\limits_{i = 1 .. k} \| \mathbf q, \mathbf c_{ji} \|$
- $\text{res} = \text{res} \cup \big\{ \text{cluster}(\mathbf c_{ji^*}) \big\}$

- return $\text{res}$

Multi-Probing for K-Means LSH:

- fix $m_p$ the number of buckets we want to retrieve
- for each $L$ hash functions
- select $m_p$ closets centroids
- then return all vectors from these $m_p$ centroids

Idea:

- a variation of Query-Adaptive LSH for K-Means LSH
- instead of a single k-means per hash maintain a pool of independent clustering results
- at the query time select the best one from the pool

Usual Query-Adaptive LSH:

- define a pool of $L$ hash functions (with $L$ larger than in usual LSHs)
- compute relevance criteria $\lambda_j$ for each $g_j$: this criteria identifies the hash functions that are more likely to return the NNs
- relevance could be: distance between the query and the center of the cell

Query-Adaptive K-Means LSH:

- $\lambda_j(\mathbf q) = \min_{i = 1..k} \| \mathbf q, \mathbf c_{ji} \|$
- and $\lambda_j(\mathbf q)$ is actually a by-product of finding the nearest centroid
- so rank $g_j(\mathbf q)$ by $\lambda_j(\mathbf q)$ then pick $p$ best and use them

Notes:

- for this to be useful need $L$ larger than usual
- it gives better performance, but it becomes more computationally expensive - may be not very critical as it's done offline

Can use Approximate K-Means or Mini-Batch K-Means

- Paulevé, Loïc, et al. "Locality sensitive hashing: A comparison of hash function types and querying mechanisms." 2010. [1]