Coresets for k-Segmentation of Streaming Data
Life-logging video streams, financial time series, and Twitter tweets are a few examples of high-dimensional signals over practically unbounded time. We consider the problem of computing optimal segmentation of such signals by a k-piecewise linear function, using only one pass over the data by maintaining a coreset for the signal. The coreset enables fast further analysis such as automatic summarization and analysis of such signals. A coreset (core-set) is a compact representation of the data seen so far, which approximates the data well for a specific task -in our case, segmentation of the stream. We show that, perhaps surprisingly, the segmentation problem admits coresets of cardinality only linear in the number of segments k, independently of both the dimension d of the signal, and its number n of points. More precisely, we construct a representation of size O(k log n/ε 2 ) that provides a (1 + ε)-approximation for the sum of squared distances to any given k-piecewise linear function. Moreover, such coresets can be constructed in a parallel streaming approach. Our results rely on a novel reduction of statistical estimations to problems in computational geometry. We empirically evaluate our algorithms on very large synthetic and real data sets from GPS, video and financial domains, using 255 machines in Amazon cloud.
The authors study the problem of k-segmentation of data and provide a compact coreset that is linear in size of the dimensions and number of segments. Strengths:1. Compact coreset that is computed in time linear in n, d, k.2. The paper is well written and provides a good summarization of existing techniques.3. The experiments include a collection of very interesting data sets. Concerns:(Note: A few concerns raised are directly in light of the work in [12], especially since the authors address [12] as the most recent paper worthy of comparison.)1. In the abstract, the authors claim that the size of the coreset is independant of d and n, but in Page 3, the size of the coreset is given as O(dk/\eps^2) and in Page 5, the size of the coreset is given as O(klog(n)/\eps^2). It is therefore very unclear what the size of the coreset is.2. The authors claim that the the size of their coreset is an improvement over [12] whose runtime is cited to be cubic in both d and k. But, the size of the coreset as claimed by [12] is linear in k.3. The cost term involving the coreset is a little convoluted in the paper involing sum of two terms, whereas [12] provides a cleaner cost function very similar to the original cost function used on the entire data. This makes it difficult to evaluate the effectiveness of the coreset.4. While the deterministic nature of the coreset computation is appealing, it is unclear how to run parallel and streaming versions of the algorithm since the effect of an adversary is scrambling the inputs each time is never discussed. 5. In its current form, the meat of the paper only has brief proof sketches and no analysis of run times making it contribution limited. The technical contribution of the paper is limited to a page since a flavor of BICRITERIA algorithm is already discussed in [12] (and not mentioned in this paper).6. In page 7, the authors claim that they significantly outperform other schemes (Figure 2), while the other schemes discussed are merely uniform coreset and a random coreset (which take constant time to compute and are therefore much faster). The authors fail to discuss and compare their approach to existing state of the art methods including [12]. The methods that they compare to are from 40 years ago.