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Property Testing of Data Dimensionality Robert Krauthgamer ICSI and UC Berkeley Joint work with Ori Sasson (Hebrew U.)

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Testing Data Dimensionality2 Data dimensionality The analysis of large volumes of complex data is required in many disciplines. Such data is frequently represented by vectors in a high- dimensional vector space. – E.g., sequential biological data (genome, proteins) – A common method of representing data is feature extraction (vector representation in feature space). Images databases Text corpora (via latent semantic indexing)

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Testing Data Dimensionality3 The issue of dimension High-dimensional data is difficult to work with. – Complexity of many operations is heavily dependent (e.g. exponentially) on the dimension. Real-life data often adheres to a low-dimensional structure – Which allows to effectively reduce the dimension. – E.g. in R 2 : Dimensionality Reduction: Mapping into low-dimensional space (while preserving most of the data “structure”) – Trade-off accuracy for computational efficiency

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Testing Data Dimensionality4 Dimensionality reduction methods Singular Value Decomposition (SVD) – I.e., low-rank matrix approximation. – Practical variants: Multidimensional Scaling (MDS), Principal Component Analysis (PCA) Low-distortion embedding in low-dimensional l p – Of any Euclidean metric [Johnson-Lindenstrauss’86] – Of any metric [Bourgain’86, Linial-London-Rabinovich’93]. Other methods, e.g. combinatorial feature selection [Charikar-Guruswami-Kumar-Rajagopalan-Sahai’00] Linear Structure Metric Structure

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Testing Data Dimensionality5 Property testing framework Relaxed decision problems: Determine whether The input has a property P, or The input is far from having the property P, i.e. it needs to be modified significantly in order to have the property. Goal: Obtain Randomized algorithms (correct with probability 2/3), Whose complexity is low (does not depend on input size). Trivial example: Testing if an input list contains only 0’s or -fraction of the entries are not 0 – with queries.

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Testing Data Dimensionality6 Testing data dimensionality Given a data set S, determine whether S has at most a (fixed) dimension d, or S is -far from having this property, – i.e. at least an -fraction of the entries of (a representation of S) needs to be modified for S to have the property. Technicalities: Interpretation of dimension (i.e. type of structure) Representation of S – Assume it affects both query mechanism and farness measure

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Testing Data Dimensionality7 Our results – Testing for linear structure Algorithm for testing whether vectors v 1,…,v n lie in linear (or affine) subspace of dimension d. – Algorithm queries O(d/ ) vectors. – Holds for every vector space V. Algorithm for testing whether a matrix A m n has rank d. – Algorithm queries the entries of an O(d/ ) O(d/ ) submatrix. – Holds for matrices over any field F. (Both algorithms have one-sided error.)

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Testing Data Dimensionality8 Our results – Testing for metric structure Testing whether v 1,…,v n l 2 m can be embedded into l 2 d – Isometrically - achieved by querying O(d/ ) vectors (corollary). – With distortion - requires querying ((n/ ) 1/2 ) vectors. – With perturbation - requires (min{n 1/2, m/log m}) queries. Testing whether vectors v 1,…,v n l 1 m can be embedded isometrically into l 1 d requires querying (n 1/4 ) vectors. (Lower bounds are for algorithms with two-sided error.)

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Testing Data Dimensionality9 Our results – Testing metrics and norm Algorithm for testing whether a matrix M n n is the distances matrix of a d-dimensional Euclidean metric. – Algorithm queries the entries of an O(d/ ) O(d/ ) submatrix. – Slight improvement over O((dlog d)/ ) O((dlog d)/ ) of [Parnas-Ron’01]. Algorithm for testing whether a vector has l p -norm . – Algorithm queries O( log 1/ ) entries (with two-sided error). – Holds for any p and . – Allows to test the Frobenius norm of a matrix (such as the difference between a matrix and its low-rank approximation).

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Testing Data Dimensionality10 Property testing origins Introduced by [Rubinfeld-Sudan’96] – Testing algebraic properties of functions Many PCPs involve testing of encodings – E.g. low-degree polynomials, Hadamard code, long code Testing of combinatorial properties initiated by [Goldreich-Goldwasser-Ron’98] – They focused on graph properties (e.g. coloring). – Later works considered testing monotonicity of functions, satisfiability of formulas, regularity of languages, equality of distributions, clustering of Euclidean vectors, metric spaces etc.

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Testing Data Dimensionality11 Related work Property testing – Testing whether a distances matrix represents a tree metric, ultra- metric, or a low-dimensional Euclidean metric [Parnas-Ron’01]. – Testing properties of Euclidean vectors, e.g. clustering [Alon- Dar-Parnas-Ron’00] and convexity [Czumaj-Sohler-Ziegler’00]. – Testing various matrix properties, e.g. monotonicity [Newman- Fischer’01]. Fast low-rank approximation (by sampling) – [Frieze-Kannan-Vempala’98, Achlioptas-McSherry’01] – Farness measure considers the magnitude of the changes. – Sampling depends on input size (unless input is “uniform”).

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Testing Data Dimensionality12 Other related work Finite point criterion for l p d – embeddability. – Namely, the minimum f p (d) such that (any) metric space embeds in l p d iff every f p (d) of its points do. – For p = 2, [Menger’28] showed f p (d) = d+3. – For p = 1 and any d > 2, [Bandelt-Chepoi-Laurent’98] showed f 1 (d) d 2 -1, but it is not known whether f 1 (d) is finite. Our results for l 1 and l 2 spaces establish somewhat similar bounds for a relaxed version of this question.

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Testing Data Dimensionality13 Algorithm for testing linear structure Thm 1. Testing whether a set of vectors S lies in a subspace of dimension d can be achieved with O(d/ ) queries. The algorithm. 1. Query O(d/ ) vectors of S uniformly at random. 2. Accept if (and only if) the queried vectors lie in a linear (or affine) subspace of dimension d.

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Testing Data Dimensionality14 Proof of testing linear structure Proof (correctness). Algorithm always accepts a data set S of dimension d. Let S be -far from having dimension d. Consider sampling the O(d/ ) vectors one by one. Let X t be the dimension of the subspace spanned by the first t sampled vectors. Lemma 1. Pr[X t+1 = X t + 1 | X t d] . Proof. Since S is -far from having dimension d, the subspace spanned by the first t sampled vectors contains less than -fraction of the vectors of S.

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Testing Data Dimensionality15 A technical lemma Lemma 2. Let 0 X 0 X 1 X 2 . be random variables. If Pr[X t+1 = X t + 1 | X t d] for all t 0, then for t* = 8d/ we have Pr[X t* d] < 1/3. Proof sketch. X t has binomial distribution as long as X t d. Then E[X t* ] 8d and using Chernoff Pr[X t* d] < 1/3. So with probability 2/3 we have X t* d and the algorithm rejects (for S that is -far from dimension d). This completes the proof of Thm 1. – Similar approach allows to test if a matrix is low-rank and for distances matrix (slight improvement over [Parnas-Ron’01]).

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Testing Data Dimensionality16 Lower bound for l 1 Thm 2. Testing whether n vectors in l 1 m can be embedded isometrically into l 1 d requires querying (n 1/4 ) vectors. Consider first algorithms with one-sided error. Suppose d=1, m=2. Consider the following point set S: S is 1/24-far from l 1 d -embeddability because every “ ” cannot be embedded in the line.

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Testing Data Dimensionality17 Lower bound for l 1 with one-sided error Assume there is an algorithm that queries t << n 1/2 points. WLOG it sees a “random” sample of S. With high probability 1 – O(t 2 /n) = 1 – o(1) – The sample contains no two points at distance O(1) from each other. – Then sample is l 1 d –embeddable (since there is a geodesic line going through all its points). – And so algorithm must accept S. Contradiction (since S is 1/24-far).

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Testing Data Dimensionality18 Lower bound for l 1 with two-sided error We (randomly) create from S another data set S’ such that – S’ embeds in the line (WHP 1-o(1)). – The algorithm’s view of S differs from its view of S’ with probability o(1), – So probabilities of accepting S vs. that of S’ differ by o(1)<<1/3. Contradiction. Here (to prove Thm 2): – Create S’ by choosing r << n 1/2 random points from S and duplicating each one n/r times. – Then a sample of << r 1/2 points from S,S’ is almost the same. These inputs look the same

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Testing Data Dimensionality19 Lower bound for l 2 with perturbation Thm 3. Testing whether n vectors in l 2 m can be perturbed by to be l 2 d – embeddable requires (min{n 1/2, m/log m}) queries. Let d=0 (I.e. testing if the vectors are in a ball of radius ). Consider a sphere of radius ’ = (1+1/2n) in l 2 m. Let S’ consist of n random vectors from this sphere. Let S consist of n/2 random vectors from the sphere and their n/2 antipodal vectors (-v).

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Testing Data Dimensionality20 Lower bound for l 2 with perturbation WHP, the vectors of S’ are in a ball of radius – By concentration of measure, WHP they are nearly orthogonal, e.g. the distance between every two is roughly . – In fact, WHP they are all at distance < from their “center of mass”, as claimed. S’ Concentration of measure YES

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Testing Data Dimensionality21 Lower bound for l 2 with perturbation S is 1/2-far from being in a ball of radius – Because the distance between antipodal vectors in S is 2 ’ . Assume algorithm queries << n 1/2 – WHP view of S, S’ is the same. – So, probability of accepting S and S’ should differ by o(1). – Contradiction. This proves Thm 3. S Antipodals NO

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Testing Data Dimensionality22 Lower bound for l 2 with distortion Thm 4: Testing whether n vectors in l 2 m can be embedded in l 2 d with distortion requires ((n/ ) 1/2 ) queries. Let d=1 (embedding into a line with distortion ). Consider a unit circle with equally spaced 10 points. Let S consist of points from n/10 (far apart) parallel copies of this circle in R 3.

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Testing Data Dimensionality23 Lower bound for l 2 with distortion S is 1/10 -far from having an embedding with distortion – Since embedding each cycle into the line requires distortion > . points NO

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Testing Data Dimensionality24 Lower bound for one-sided error Assume algorithm queries << (n/ ) 1/2 points of S – WLOG it sees a “random” sample of S. – WHP, this sample contains at most one point from each circle, – And then it can be embedded with distortion < into the line (by mapping each point to its circle’s center). – So WHP algorithm must accept S. Contradiction. points YES

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Testing Data Dimensionality25 Lower bound for two-sided error We create S’ by choosing one point from each circle of S and duplicating it 10 times. – Then S’ can be embedded with distortion < into the line. – WHP view of << (n/ ) 1/2 points from S is the same as from S’. – So, probability of accepting S and S’ should differ by o(1). – This proves Thm 4.

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Testing Data Dimensionality26 Future research Testing whether – A matrix spectral norm ||A|| 2 is small. – A distances matrix represents metric (triangle inequality). – A distances matrix represents an l 1 d – metric. – A distances matrix represents an approximate l 2 d – metric. Testing with farness measure that depends on magnitude – a la [Frieze-Kannan-Vempala’98, Achlioptas-McSherry’01]

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