Sparser Johnson-Lindenstrauss Transforms

We give two different and simple constructions for dimensionality reduction in $\ell_2$ via linear mappings that are sparse: only an $O(\varepsilon)$-fraction of entries in each column of our embedding matrices are non-zero to achieve distortion $1+\varepsilon$ with high probability, while still achieving the asymptotically optimal number of rows. These are the first constructions to provide subconstant sparsity for all values of parameters, improving upon previous works of Achlioptas (JCSS 2003) and Dasgupta, Kumar, and Sarl\'{o}s (STOC 2010). Such distributions can be used to speed up applications where $\ell_2$ dimensionality reduction is used.Comment: v6: journal version, minor changes, added Remark 23; v5: modified abstract, fixed typos, added open problem section; v4: simplified section 4 by giving 1 analysis that covers both constructions; v3: proof of Theorem 25 in v2 was written incorrectly, now fixed; v2: Added another construction achieving same upper bound, and added proof of near-tight lower bound for DKS schem

Bounded Independence Fools Degree-2 Threshold Functions

Let x be a random vector coming from any k-wise independent distribution over {-1,1}^n. For an n-variate degree-2 polynomial p, we prove that E[sgn(p(x))] is determined up to an additive epsilon for k = poly(1/epsilon). This answers an open question of Diakonikolas et al. (FOCS 2009). Using standard constructions of k-wise independent distributions, we obtain a broad class of explicit generators that epsilon-fool the class of degree-2 threshold functions with seed length log(n)*poly(1/epsilon). Our approach is quite robust: it easily extends to yield that the intersection of any constant number of degree-2 threshold functions is epsilon-fooled by poly(1/epsilon)-wise independence. Our results also hold if the entries of x are k-wise independent standard normals, implying for example that bounded independence derandomizes the Goemans-Williamson hyperplane rounding scheme. To achieve our results, we introduce a technique we dub multivariate FT-mollification, a generalization of the univariate form introduced by Kane et al. (SODA 2010) in the context of streaming algorithms. Along the way we prove a generalized hypercontractive inequality for quadratic forms which takes the operator norm of the associated matrix into account. These techniques may be of independent interest.Comment: Using v1 numbering: removed Lemma G.5 from the Appendix (it was wrong). Net effect is that Theorem G.6 reduces the m^6 dependence of Theorem 8.1 to m^4, not m^

Bounded Independence Fools Degree-2 Threshold Functions

For an n-variate degree-2 real polynomial p, we prove that $E_{x\sim D}[sig(p(x))]$ Is determined up to an additive $\epsilon$ as long as D is a k-wise Independent distribution over $\{-1, 1\}^n$ for $k = poly(1/\epsilon)$. This gives a broad class of explicit pseudorandom generators against degree-2 boolean threshold functions, and answers an open question of Diakonikolas et al. (FOCS 2009).Engineering and Applied Science

On the Exact Space Complexity of Sketching and Streaming Small Norms

We settle the 1-pass space complexity of $(1 \pm \epsilon)$-approximating the $L_p$ norm, for real p with 1 ≤ p ≤ 2, of a length-n vector updated in a length-m stream with updates to its coordinates. We assume the updates are integers in the range [–M, M]. In particular, we show the space required is $\Theta(\epsilon^{−2} log(mM) + log log(n))$ bits. Our result also holds for 0 < p < 1; although $L_p$ is not a norm in this case, it remains a well-defined function. Our upper bound improves upon previous algorithms of [Indyk, JACM ‘06] and [Li, SODA ‘08]. This improvement comes from showing an improved derandomization of the $L_p$ sketch of Indyk by using k-wise independence for small k, as opposed to using the heavy hammer of a generic pseudorandom generator against space-bounded computation such as Nisan's PRG. Our lower bound improves upon previous work of [Alon-Matias-Szegedy, JCSS ‘99] and [Woodruff, SODA ‘04], and is based on showing a direct sum property for the 1-way communication of the gap-Hamming problem.Engineering and Applied Science

An Optimal Algorithm for the Distinct Elements Problem

We give the first optimal algorithm for estimating the number of distinct elements in a data stream, closing a long line of theoretical research on this problem begun by Flajolet and Martin in their seminal paper in FOCS 1983. This problem has applications to query optimization, Internet routing, network topology, and data mining. For a stream of indices in {1,...,n}, our algorithm computes a $(1 \pm \epsilon)$-approximation using an optimal $O(1/\epsilon^{-2} + log(n))$ bits of space with 2/3 success probability, where $0<\epsilon<1$ is given. This probability can be amplified by independent repetition. Furthermore, our algorithm processes each stream update in O(1) worst-case time, and can report an estimate at any point midstream in O(1) worst-case time, thus settling both the space and time complexities simultaneously. We also give an algorithm to estimate the Hamming norm of a stream, a generalization of the number of distinct elements, which is useful in data cleaning, packet tracing, and database auditing. Our algorithm uses nearly optimal space, and has optimal O(1) update and reporting times.Engineering and Applied Science

Fast Moment Estimation in Data Streams in Optimal Space

We give a space-optimal streaming algorithm with update time $O(log^2(1/\epsilon)loglog(1/\epsilon))$ for approximating the pth frequency moment, 0 < p < 2, of a length-n vector updated in a data stream up to a factor of $1 \pm \epsilon$. This provides a nearly exponential improvement over the previous space optimal algorithm of [Kane-Nelson-Woodruff, SODA 2010], which had update time $\Omega(1/\epsilon^2)$. When combined with the work of [Harvey-Nelson-Onak, FOCS 2008], we also obtain the first algorithm for entropy estimation in turnstile streams which simultaneously achieves near-optimal space and fast update time.Engineering and Applied Science

FUTURE CARDIAC EVENTS IN NORMALLY DIAGNOSED GATED MYOCARDIAL PERFUSION SPECT (GSPECT)

Coronary heart disease (CHD) is a major cause of mortality and morbidity in Europe and USA and its management consumes a large proportion of national healthcare budgets. Many studies had tested the prognostic value of a normal myocardial perfusion scintigraphy; they concluded that a normal MPI study is associated with a very low rate of future cardiac events. In view of the above this study is designed to determine the risk of future cardiac events after normal MPS in local population. Methods: This was a retrospective observational registry performed in a single center in the Kingdom of Saudi Arabia. The data were collected from the nuclear medicine database identifying all the reported normal myocardial perfusion scans between January 2008 and December 2011 . Results: There were 290 patients identified with normal cardiac nuclear scans in the pre-specified time frame. Basic patient demographics were outlined and the patients’ charts were reviewed looking for any major cardiac events such as MI or sudden death. Mean follow up was 14.8 months. There were 2 patients that were admitted with NSTEMI and went on to have diagnostic angiograms. One of those two patients underwent percutaneous coronary intervention with stenting. The other patient had nonobstructive CAD and was advised for medical management only. These findings are consistent with a 0.7% risk of cardiac events after a negative scan Conclusion: The above findings demonstrate that the risk of major cardiac events after a negative nuclear cardiac scan is low and is in keeping with the international statistics available

FUTURE CARDIAC EVENTS IN NORMALLY DIAGNOSED GATED MYOCARDIAL PERFUSION SPECT (GSPECT)

Coronary heart disease (CHD) is a major cause of mortality and morbidity in Europe and USA and its management consumes a large proportion of national healthcare budgets. Many studies had tested the prognostic value of a normal myocardial perfusion scintigraphy; they concluded that a normal MPI study is associated with a very low rate of future cardiac events. In view of the above this study is designed to determine the risk of future cardiac events after normal MPS in local population. Methods: This was a retrospective observational registry performed in a single center in the Kingdom of Saudi Arabia. The data were collected from the nuclear medicine database identifying all the reported normal myocardial perfusion scans between January 2008 and December 2011 . Results: There were 290 patients identified with normal cardiac nuclear scans in the pre-specified time frame. Basic patient demographics were outlined and the patients’ charts were reviewed looking for any major cardiac events such as MI or sudden death. Mean follow up was 14.8 months. There were 2 patients that were admitted with NSTEMI and went on to have diagnostic angiograms. One of those two patients underwent percutaneous coronary intervention with stenting. The other patient had nonobstructive CAD and was advised for medical management only. These findings are consistent with a 0.7% risk of cardiac events after a negative scan Conclusion: The above findings demonstrate that the risk of major cardiac events after a negative nuclear cardiac scan is low and is in keeping with the international statistics available