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Sebastian Bruch
# Foundations of Vector Retrieval
# Preface
We are witness to a few years of remarkable developments in Artificial Intelligence with the use of advanced machine learning algorithms, and in particular, deep learning. Gargantuan, complex ... | Foundations of Vector Retrieval | https://arxiv.org/abs/2401.09350 | Foundations of Vector Retrieval arXiv:2401.09350 | [
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1 | These neural networks and their training algorithms may be complex, and the scope of their impact broad and wide, but nonetheless they are simply functions in a high-dimensional space. A trained neural network takes a vector as input, crunches and transforms it in various ways, and produces another vector, often in som... | Foundations of Vector Retrieval | https://arxiv.org/abs/2401.09350 | Foundations of Vector Retrieval arXiv:2401.09350 | [
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2 | If new and old knowledge can be squeezed into a collection of learnt or hand-crafted vectors, what useful things does that enable us to do? A metaphor that might help us think about that question is this: An ever- evolving database full of such vectors that capture various pieces of data can
v
# vi
be understood as a m... | Foundations of Vector Retrieval | https://arxiv.org/abs/2401.09350 | Foundations of Vector Retrieval arXiv:2401.09350 | [
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3 | Similarity is then a function of two vectors, quantifying how similar two vectors are. It may, for example, be based on the Euclidean distance between the query vector and a database vector, where similar vectors have a smaller distance. Or it may instead be based on the inner product between two vec- tors. Or their an... | Foundations of Vector Retrieval | https://arxiv.org/abs/2401.09350 | Foundations of Vector Retrieval arXiv:2401.09350 | [
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4 | A neural network that is trained to perform a general task such as question- answering, could conceivably augment its view of the world by ârecallingâ in- formation from such a database and finding answers to new questions. This is particularly useful for generative agents such as chatbots who would oth- erwise be ... | Foundations of Vector Retrieval | https://arxiv.org/abs/2401.09350 | Foundations of Vector Retrieval arXiv:2401.09350 | [
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5 | For decades now, research on vector retrieval has sought to improve the efficiency of search over large vector databases. The resulting literature is rich with solutions ranging from heavily theoretical results to performant empir- ical heuristics. Many of the proposed algorithms have undergone rigorous benchmarking an... | Foundations of Vector Retrieval | https://arxiv.org/abs/2401.09350 | Foundations of Vector Retrieval arXiv:2401.09350 | [
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6 | That gap is what this monograph intends to close. With the goal of present- ing the fundamentals of vector retrieval as a sub-discipline, this manuscript delves into important data structures and algorithms that have emerged in the literature to solve the vector retrieval problem efficiently and effectively.
vii
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7 | # Retrieval Algorithms
With that foundation in place and the question clearly formulated, the second part of the monograph explores the different classes of existing solutions in great depth. We close each chapter with a summary of algorithmic insights. There, we will also discuss what remains challenging and explore f... | Foundations of Vector Retrieval | https://arxiv.org/abs/2401.09350 | Foundations of Vector Retrieval arXiv:2401.09350 | [
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8 | Alternatively, instead of laying a mesh over the space, we may define a fixed number of buckets and map data points to these buckets with the property that, if two data points are close to each other according to the distance func- tion, they are more likely to be mapped to the same bucket. When processing a query, we ... | Foundations of Vector Retrieval | https://arxiv.org/abs/2401.09350 | Foundations of Vector Retrieval arXiv:2401.09350 | [
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9 | As we examine vector retrieval algorithms, it is inevitable that we must ink in extra pages to discuss why similarity based on inner product is special and why it poses extra challenges for the algorithms in each categoryâmany of these difficulties will become clear in the introductory chapters.
There is, however, a ... | Foundations of Vector Retrieval | https://arxiv.org/abs/2401.09350 | Foundations of Vector Retrieval arXiv:2401.09350 | [
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10 | Related to the topic of compression is the concept of sketching. Sketching is a technique to project a high-dimensional vector into a low-dimensional vector, called a sketch, such that certain properties (e.g., the L2 norm, or inner products between any two vectors) are approximately preserved. This probabilistic metho... | Foundations of Vector Retrieval | https://arxiv.org/abs/2401.09350 | Foundations of Vector Retrieval arXiv:2401.09350 | [
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11 | # Intended Audience
This monograph is intended as an introductory text for graduate students who wish to embark on research on vector retrieval. It is also meant to serve as a self-contained reference that captures important developments in the field, and as such, may be useful to established researchers as well.
As th... | Foundations of Vector Retrieval | https://arxiv.org/abs/2401.09350 | Foundations of Vector Retrieval arXiv:2401.09350 | [
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12 | # Acknowledgements
I am forever indebted to my dearest colleagues Edo Liberty, Amir Ingber, Brian Hentschel, and Aditya Krishnan. This incredible but humble group of scholars at Pinecone are generous with their time and knowledge, patiently teaching me what I do not know, and letting me use them as a sounding board wit... | Foundations of Vector Retrieval | https://arxiv.org/abs/2401.09350 | Foundations of Vector Retrieval arXiv:2401.09350 | [
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13 | xi
# Notation
This section summarizes the special symbols and notation used throughout this work. We often repeat these definitions in context as a reminder, espe- cially if we choose to abuse notation for brevity or other reasons.
Paragraphs that are highlighted in a gray box such as this contain important statements,... | Foundations of Vector Retrieval | https://arxiv.org/abs/2401.09350 | Foundations of Vector Retrieval arXiv:2401.09350 | [
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14 | # Sets
J |·| [n] B(u, r) \ â³ 1p Calligraphic font typically denotes sets. The cardinality (number of items) of a finite set. The {1, 2, 3, . . . , n}. The closed ball of radius r centered at point u: {v | δ(u, v) ⤠r} where δ(·, ·) is the distance function. The set difference operator: A \ B = {x â A | x /â... | Foundations of Vector Retrieval | https://arxiv.org/abs/2401.09350 | Foundations of Vector Retrieval arXiv:2401.09350 | [
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15 | Z The set of integers.
R¢ d-dimensional Euclidean space.
g@-1 The hypersphere in R¢.
U,U,W Lowercase letters denote vectors.
Ui, Vi, Wi Subscripts identify a specific coordinate of a vector, so that u,; is the i-th coordinate of vector u.
# Functions and Operators
nz (·)
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17 | Intrinsic Dimensionality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.1 High-Dimensional Data and Low-Dimensional Manifolds . . . . . 23 3.2 Doubling Measure and Expansion Rate . . . . . . . . . . . . . . . . . . . . 24 3.3 Doubling Dimension . . . . . . . . . . . . . . . . . . . . . . . . .... | Foundations of Vector Retrieval | https://arxiv.org/abs/2401.09350 | Foundations of Vector Retrieval arXiv:2401.09350 | [
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18 | 1 Vector Retrieval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1 Vector Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Vectors as Units of Retrieval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 . . . . . . . . . . . . ... | Foundations of Vector Retrieval | https://arxiv.org/abs/2401.09350 | Foundations of Vector Retrieval arXiv:2401.09350 | [
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20 | 2 Retrieval Stability in High Dimensions . . . . . . . . . . . . . . . . . . . 17 2.1 Intuition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2 Formal Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.3 Empirical Demonstrati... | Foundations of Vector Retrieval | https://arxiv.org/abs/2401.09350 | Foundations of Vector Retrieval arXiv:2401.09350 | [
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21 | Locality Sensitive Hashing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 5.1 Intuition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 5.2 Top-k Retrieval with LSH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 5.2.1 The Point Locatio... | Foundations of Vector Retrieval | https://arxiv.org/abs/2401.09350 | Foundations of Vector Retrieval arXiv:2401.09350 | [
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22 | . . . . . . . . . . . . . . . . . 63 5.3.2 Angular Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 5.3.3 Euclidean Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 5.3.4 Inner Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 5.4 Cl... | Foundations of Vector Retrieval | https://arxiv.org/abs/2401.09350 | Foundations of Vector Retrieval arXiv:2401.09350 | [
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