CS276A Practice Problem Set 1 Solutions

Transcription

1 CS76A Practice Problem Set Solutios Problem. (i) (ii) 8 (iii) 6 Compute the gamma-codes for the followig itegers: (i) (ii) 8 (iii) 6 Problem. For this problem, we will be dealig with a collectio of millio documets cotaiig 5K distict terms. Assume that term occurreces follow the Zipf distributio ad that the gaps for each term are uiformly distributed (as doe i class). Estimate the size of the etire postigs file uder the followig coditios (i all cases, the postigs are stored usig gaps): (i) Uary-ecoded postigs lists (ii) Gamma-ecoded postigs lists (ii) Repeat (ii) whe the most frequet words are ot stored i the idex. As i the lecture otes, assume that the most frequet term has a postigs list with gaps of each, the secod most frequet with gaps of each, etc. Here, = millio ad the umber of distict terms m = 5K. (i) The postigs list for the k th most frequet term takes k k = bits. Hece total size is m GB. (ii) The postigs list for the k th most frequet term takes k ( log k + ) bits. Hece, we eed to compute 5K k= k ( log k + ) = < 9 i= 9 i= 9 = k< i i= 9 k= i k ( log k + ) [(i ) + ] [(i ) + ] i= i = 6 bits 9MB 9 i= k< i k k= i { i i } i assumed that k = i, for a upper boud

2 (iii) We ow eed to compute 5K k= k ( log k + ) < 9 k< i 9 k= i k ( log k + ) i = bits 8MB 9 Problem. Usig the search egie of your choice, fid a olie Porter s stemmig tool, ad stem the followig words: (i) automobile (ii) automotive (iii) cars (iv) iformatio (v) iformative We used the stemmer at Term automobile automotive cars iformatio iformative Stem automobil automot car iform iform Problem. I a iverted idex over.5 millio documets, the followig term-frequecy statistics were observed: Term Documet frequecy eyes kaleidoscope 879 marmalade 79 skies 7658 tagerie 665 trees 68 Recommed a query processig order for the followig queries: (i) (tagerie OR trees) AND (marmalade OR skies) AND (kaleidoscope OR eyes) (ii) tagerie AND (NOT marmalade) AND (NOT trees)

3 Recall the query processig heuristic of choosig terms i icreasig order of their frequecy whe computig a AND query. The frequecy of a OR expressio is estimated to be the sum of the frequecy of its idividual compoets. I additio, for a term t, NOT t has a associated frequecy of N frequecy(t), where N is the total umber of documets. Usig these rules: (i) ((kaleidoscope OR eyes) AND (tagerie OR trees)) AND (marmalade or skies) (ii) (tagerie AND (NOT trees)) AND (NOT marmalade) Problem 5. Give the followig fragmet from a iverted idex (augmeted with positioal iformatio), state how ofte the phrase The udiscovered coutry appears i each documet. the: :,8,55; 5:,6,5,; 7:67,87,9,; :,9,5,6; udiscovered: :,5,9; 5:,5,7,,5,96; 6:,5,55,6; 7:,68,7,85,; :5,,,65,8; coutry: :,6; 5:8,6,5,65; 7:5,69,9,5; 8:,,65,9; :,,75,8; The occurreces of The udiscovered coutry are listed i the followig table. Documet Occurreces 5 at (6,7,8) ad (,5,6) 7 at (67,68,69) Problem 6. Give a example that illustrates how the term-wise groupig techique may fail to accurately retrieve the top k documets for a query. Cosider a collectio with the followig ormalized documet vectors: d = (.6,,...) d = (.5,,...) d = (,.5,...)

4 d = (,.,...) d 5 = (.,.,...) Let t... t m be the set of terms i the collectio ad assume that the i th compoet of each vector correspods to term t i. Let us apply the term-wise groupig techique with k = for the query q = t t. Clearly, the preferred list for t is {d, d } ad for t is {d, d }. Hece, the cadidate set for the query is {d, d, d, d }. Now, sice t ad t each occur i documets, their IDFs are the same ad the ormalized query vector is therefore (, ). Computig cosies, we see that cosie(q, d ) =.6, cosie(q, d ) = cosie(q, d ) =.5, cosie(q, d ) =., ad cosie(q, d 5 ) =.7. Therefore, d 5 is the top-raked documet for query q but is outside the cadidate set specified by the term-wise groupig techique. Problem 7. Give a example that illustrates how the samplig ad pre-groupig techique may fail to accurately retrieve the top k documets for a query. There are of course may possible aswers. Oe example (-d case): assume the documets of the corpus are uiformly distributed o the first quadrat of the uit circle. If the chose leaders are (,) ad (,), the query poits lyig ear (.77,.77) will lead to poor results. Problem 8. This problem deals with the vector space model (usig TF-IDF weights) as applied to the followig collectio of documets: Doc : Iformatio Retrieval Systems Doc : Iformatio Storage Doc : Digital Speech Sythesis Systems Doc : Speech Filterig, Speech Retrieval (i) Compute all o-zero etries i the ormalized vector for Doc. (ii) Rak all the documets i the collectio for the query Speech Systems? (iii) Compute the cosie similarities betwee (a) docs ad (b) docs ad. The table below summarizes the relevat TF ad IDF iformatio for the etire collectio. For all computatios below, we shall use log base. TF Term IDF Doc Doc Doc Doc Digital log Filterig log Iformatio log Retrieval log Speech log Storage log Sythesis log Systems log

5 (i) Multiplyig the IDF colum with the TF colum for Doc yields the vector (,,,,,,, ) which, upo ormalizatio results i (,,,,,,, (ii) Treatig the query Speech Systems as a documet, the ormalized query vector computes to (,,,,,,, ). Therefore, cosie(q, doc) = 6, cosie(q, doc) =, cosie(q, doc) = ad cosie(q, doc) = 9. Hece, the documets i decreasig order of rak are: Doc, Doc, Doc, Doc. (iii) (a) 5 (b) 5 ). 5, Problem 9. This problem will try to give more isight ito LSI. Cosider the equatio: A s = W S s V T (i) What are the oly useful colums of W, ad the useful rows of V T (equivaletly, the useful colums of V ) (Cosider W S s ad S s V T i tur)? The first s colums of W, ad the first s rows of V T (i.e., the first s colums of V ) are the oly oes relevat to the product. (ii) What is special about the vector space spaed by these colums of W? The first s colums of W spa the same vector space as the colums of A s. The first s colums of W thus form a orthoormal basis for the documet vectors represeted by the colums of A s. (iii) The colums of W are orthoormal. Usig this fact, ad the aswer to part (ii), what does each colum of S s V T represet? A little thought will show that the colum j of A s is give by the product of W, S s, ad the j th colum of V T. Each colum of S s V T represets a documet vector, i a s-dimesioal vector space with coordiates i terms of the basis give by the first s colums of W. (iv) Ier products (ad thus cosie) are ivariat uder a chage of orthoormal bases. Kowig this, ad the aswers to the above, what is a efficiet way to calculate the cosie similarity betwee two documets d i ad d j? Take cosie betwee the s-dimesioal vectors give by the i th ad j th colums of S s V T, istead of betwee the correspodig m-dimesioal colums of A s, (v) Let B = S s V T. Note that B is a s matrix, with s << m. Show that the documet-documet similarity matrix A T s A s is also give by B T B. A T s A s = (W S s V T ) T (W S s V T ) = V S s W T W S s V T = V S s S s V T = (V S s )(S s V T ) = B T B 5