nips nips2008 nips2008-16 knowledge-graph by maker-knowledge-mining

16 nips-2008-Adaptive Template Matching with Shift-Invariant Semi-NMF

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Author: Jonathan L. Roux, Alain D. Cheveigné, Lucas C. Parra

Abstract: How does one extract unknown but stereotypical events that are linearly superimposed within a signal with variable latencies and variable amplitudes? One could think of using template matching or matching pursuit to ﬁnd the arbitrarily shifted linear components. However, traditional matching approaches require that the templates be known a priori. To overcome this restriction we use instead semi Non-Negative Matrix Factorization (semiNMF) that we extend to allow for time shifts when matching the templates to the signal. The algorithm estimates templates directly from the data along with their non-negative amplitudes. The resulting method can be thought of as an adaptive template matching procedure. We demonstrate the procedure on the task of extracting spikes from single channel extracellular recordings. On these data the algorithm essentially performs spike detection and unsupervised spike clustering. Results on simulated data and extracellular recordings indicate that the method performs well for signalto-noise ratios of 6dB or higher and that spike templates are recovered accurately provided they are suﬃciently diﬀerent. 1

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 edu Abstract How does one extract unknown but stereotypical events that are linearly superimposed within a signal with variable latencies and variable amplitudes? [sent-11, score-0.215]

2 One could think of using template matching or matching pursuit to ﬁnd the arbitrarily shifted linear components. [sent-12, score-0.345]

3 However, traditional matching approaches require that the templates be known a priori. [sent-13, score-0.631]

4 To overcome this restriction we use instead semi Non-Negative Matrix Factorization (semiNMF) that we extend to allow for time shifts when matching the templates to the signal. [sent-14, score-0.692]

5 The algorithm estimates templates directly from the data along with their non-negative amplitudes. [sent-15, score-0.576]

6 The resulting method can be thought of as an adaptive template matching procedure. [sent-16, score-0.217]

7 We demonstrate the procedure on the task of extracting spikes from single channel extracellular recordings. [sent-17, score-0.249]

8 On these data the algorithm essentially performs spike detection and unsupervised spike clustering. [sent-18, score-0.547]

9 Results on simulated data and extracellular recordings indicate that the method performs well for signalto-noise ratios of 6dB or higher and that spike templates are recovered accurately provided they are suﬃciently diﬀerent. [sent-19, score-1.058]

10 1 Introduction It is often the case that an observed waveform is the superposition of elementary waveforms, taken from a limited set and added with variable latencies and variable but positive amplitudes. [sent-20, score-0.249]

11 Examples are a music waveform, made up of the superposition of stereotyped instrumental notes, or extracellular recordings of nerve activity, made up of the superposition of spikes from multiple neurons. [sent-21, score-0.56]

12 Additionally, the elementary events are often temporally compact and their occurrence temporally sparse. [sent-23, score-0.229]

13 Conventional template matching uses a known template and correlates it with the signal; events are assumed to occur at times where the correlation is high. [sent-24, score-0.537]

14 Multiple template matching raises combinatorial issues that are addressed by Matching Pursuit [1]. [sent-25, score-0.217]

15 We ∗ Corresponding author wondered whether one can estimate the templates directly from the data, together with their timing and amplitude. [sent-27, score-0.606]

16 Over the last decade a number of blind decomposition methods have been developed that address a similar problem: given data, can one ﬁnd the amplitudes and proﬁles of constituent signals that explain the data in some optimal way. [sent-28, score-0.389]

17 This includes independent component analysis (ICA), non-negative matrix factorization (NMF), and a variety of other blind source separation algorithms. [sent-29, score-0.103]

18 The diﬀerent algorithms all assume a linear superposition of the templates, but vary in their speciﬁc assumptions about the statistics of the templates and the mixing process. [sent-30, score-0.657]

19 NMF constrains weights to be nonnegative but requires templates to also be non-negative. [sent-33, score-0.576]

20 We begin with the conventional formulation of the NMF modeling task as a matrix factorization problem and then derive in the subsequent section the case of a 1D sequence of data. [sent-36, score-0.103]

21 Matrix A can be thought of as component amplitudes and the rows of matrix B are the component templates. [sent-38, score-0.431]

22 Semi-NMF drops the non-negative constraint for B, while shift-NMF allows the component templates to be shifted in time. [sent-39, score-0.656]

23 ) The goal is to model this data as a linear superposition of K component templates Bkt with amplitudes Ank , i. [sent-44, score-1.046]

24 (5) In these expressions, k is a summation index; (M )−1 stands for matrix inverse of M ; and, 1 (M )+ = 2 (|M | + M ) and (M )− = 1 (|M | − M ) are to be applied on each element of matrix 2 M . [sent-51, score-0.108]

25 In the course of time, various events of unknown identity and variable amplitude appear in this signal. [sent-55, score-0.216]

26 We describe an event of type k with a template Bkl of length L. [sent-56, score-0.233]

27 An event can occur at any point in time, say at time sample n, and it may have a variable amplitude. [sent-58, score-0.121]

28 In addition, we do not know a priori what the event type is and so we assign to each time sample n and each event type k an amplitude Ank ≥ 0. [sent-59, score-0.25]

29 The goal is to ﬁnd the templates B and amplitudes A that explain the data. [sent-60, score-0.965]

30 In this formulation of the model, the timing of an event is given by a non-zero sample in the amplitude matrix A. [sent-61, score-0.225]

31 This means that for a given n the estimated amplitudes are positive for only one k, and neighboring samples in time have zero amplitudes. [sent-63, score-0.439]

32 Each block implements a convolution of the k-th template Bkl with amplitudes signal Ank . [sent-68, score-0.574]

33 We will also require a unit-norm constraint on the K templates in B, namely, l Bk Bkl = 1, to disambiguate the arbitrary scale in the product of A and B. [sent-70, score-0.615]

34 2 Optimization criterion with sparseness prior Under the assumption that the data represent a small set of well-localized events, matrix A should consist of a sparse series of pulses, the other samples having zero amplitude. [sent-72, score-0.111]

35 Assuming Gaussian white noise, the new cost function given by the negative log-posterior reads (up to a scaling factor), E = = 1 ˆ 2 ||X − X||2 + β ||A||α α 2 2 1 ˜kl Xt − At Bkl + β 2 t (10) Aα , kl (11) kl where ||·||p denotes the Lp norm (or quasi-norm for 0 < p < 1). [sent-74, score-0.102]

36 3 A update The update for A which minimizes this cost function is similar to update (5) with some modiﬁcations. [sent-79, score-0.105]

37 In (5), amplitudes A can be treated as a matrix of dimensions T × K and each update can be applied separately for every n. [sent-80, score-0.466]

38 B is now a T K × T matrix of shifted ˜ templates deﬁned as Bnkt = Bk,t−n . [sent-82, score-0.659]

39 (12) + αβAα−1 nk The summation in the BB T term is over t, and is 0 most of the time when the events do not ˜l ˜l overlap. [sent-84, score-0.184]

40 4 B update The templates B that minimize the square modeling error, i. [sent-91, score-0.611]

41 (13) The matrix inverse is now over a matrix of LK by LK elements. [sent-94, score-0.084]

42 The unit-norm constraint for the templates B therefore prevents A from shrinking arbitrarily. [sent-97, score-0.615]

43 5 Normalization The normalization constraint of the templates B can be implemented using Lagrange multipliers, leading to the constrained least squares solution: ˜kl ˜ Bkl = At Atk l + Λkl,k l −1 ˜t Ak l X t . [sent-99, score-0.615]

44 (14) Here, Λkl,k l represents a diagonal matrix of size KL × KL with K diﬀerent Lagrange l multipliers as parameters that need to be adjusted so that Bk Bkl = 1 for all k. [sent-100, score-0.084]

45 The algorithm is then applied to extracellular recordings of neuronal spiking activity and we evaluate its ability to recover two distinct spike types that are typically superimposed in this data. [sent-106, score-0.481]

46 In addition we report how accurately the templates have been recovered. [sent-115, score-0.576]

47 We generated synthetic spike trains with two types of “spikes” and added Gaussian white noise. [sent-116, score-0.315]

48 The two sets of panels show the templates B (original on the left and recovered on the right), amplitudes A (same as above) and noisy data ˆ X (left) and estimated X (right). [sent-118, score-1.04]

49 Clearly, for this SNR the templates have been recovered accurately and their occurrences within the waveform have been found with only a few missing events. [sent-120, score-0.732]

50 Detection rate is measured as the number of events recovered over the total number of events in the original data. [sent-122, score-0.319]

51 Presence or absence of a recovered event is determined by comparing the original pulse train with the reconstructed pulse train A (channel number k is ignored). [sent-124, score-0.268]

52 Templates in this example have a correlation time (3 dB down) of 2-4 samples and so we tolerate a misalignment of events of up to ±2 samples. [sent-125, score-0.16]

53 We simulated 30 events with amplitudes uniformly distributed in [0, 1]. [sent-126, score-0.523]

54 The algorithm tends to miss smaller events with amplitudes comparable to the noise amplitude. [sent-127, score-0.546]

55 To capture this eﬀect, we also report a detection rate that is weighted by event amplitude. [sent-128, score-0.13]

56 Some events may be detected but assigned to the wrong template. [sent-129, score-0.134]

57 Finally, we report the goodness of ﬁt as R2 for the templates B and the continuous valued amplitudes A for the events that are present in the original data. [sent-131, score-1.099]

58 Obviously, the performance of this type of unsupervised clustering will degrade as the templates become more and more similar. [sent-133, score-0.576]

59 Figure 2 shows the same performance numbers as a function of the similarity of the templates (without additive noise). [sent-134, score-0.625]

60 A similarity of 0 corresponds to the templates shown as examples in Figure 1 (these are almost orthogonal with a cosine of 74◦ ), and similarity 1 means identical templates. [sent-135, score-0.696]

61 Evidently the algorithm is most reliable when the target templates are dissimilar. [sent-136, score-0.576]

62 2 Analysis of extracellular recordings The original motivation for this algorithm was to analyze extracellular recordings from single electrodes in the guinea pig cochlear nucleus. [sent-138, score-0.439]

63 Spherical and globular bushy cells in the anteroventral cochlear nucleus (AVCN) are assumed to function as reliable relays of spike trains from the auditory nerve, with “primary-like” responses that resemble those of auditory nerve ﬁbers. [sent-139, score-0.541]

64 Every incoming spike evokes a discharge within the outgoing axon [6]. [sent-140, score-0.274]

65 Error bars represent standard deviation over 100 repetitions with varying random amplitudes and random noise. [sent-172, score-0.389]

66 Bottom left: false alarm rate (detected events which do not correspond to an event in the original data). [sent-176, score-0.263]

67 However, recent observations give a more nuanced picture, suggesting that the post-synaptic spike may sometimes be suppressed according to a process that is not well understood [7]. [sent-180, score-0.244]

68 Their relative amplitudes depend upon the position of the electrode tip relative to the cell. [sent-183, score-0.413]

69 Our aim is to isolate each of these components to understand the process by which the SD spike is sometimes suppressed. [sent-184, score-0.276]

70 The events may overlap in time (in particular the SD spike always overlaps with an IS spike), with varying positive amplitudes. [sent-185, score-0.404]

71 The assumptions of our algorithm are met by these data, as well as by multi-unit recordings reﬂecting the activity of several neurons (the “spike sorting problem”). [sent-187, score-0.09]

72 The automatically recovered spike templates seem to capture a number of the key features. [sent-193, score-0.871]

73 Template 1, in blue, resembles the SD spike, and template 2, in red, is similar to the IS spike. [sent-194, score-0.162]

74 The SD spikes are larger and have sharper peaks as compared to the IS spikes, while the IS spikes have an initial peak at 0. [sent-195, score-0.26]

75 The larger size of the extracted spikes corresponding to template 1 is correctly reﬂected in the histogram of the recovered amplitudes. [sent-197, score-0.343]

76 The main diﬀerence is in the small peak preceding the template 1. [sent-199, score-0.162]

77 This is perhaps to be expected as the SD spike is always preceded in the raw data by a smaller IS spike. [sent-200, score-0.244]

78 The expected templates were very similar (with a cosine of 38◦ as estimated from the manually extracted spikes), making the task particularly diﬃcult. [sent-201, score-0.646]

79 Reconstructed waveform and residual 2 Manually constructed templates B Reconstructed Residual 1 SD 2 IS mV 1 0 −1 0 0. [sent-202, score-0.704]

80 875 Time (ms) Distribution of the amplitudes A 2. [sent-210, score-0.389]

81 Top left: reconstructed waveform (blue) and residual between the original data and the reconstructed waveform (red). [sent-220, score-0.369]

82 Top right: templates B estimated manually from the data. [sent-221, score-0.624]

83 The SD spikes (blue) generally occur with larger amplitudes than the IS spikes (red). [sent-224, score-0.673]

84 In addition to these multiple restarts, we use a few heuristics that are motivated by the desired result of spike detection. [sent-228, score-0.244]

85 We can thus prevent the algorithm from converging to some obviously suboptimal solutions: Re-centering the templates: We noticed that local minima with poor performance typically occurred when the templates B were not centered within the L lags. [sent-229, score-0.576]

86 In those cases the main peaks could be adjusted to ﬁt the data, but the portion of the template that extends outside the window of L samples could not be adjusted. [sent-230, score-0.187]

87 To prune these suboptimal solutions, it was suﬃcient to center the templates during the updates while shifting the amplitudes accordingly. [sent-231, score-0.965]

88 Pruning events: We observed that spikes tended to generate non-zero amplitudes in A in clusters of 1 to 3 samples. [sent-232, score-0.519]

89 Spike amplitude was preserved by scaling the pulse amplitudes to match the sum of amplitudes in the cluster. [sent-234, score-0.899]

90 Re-training with a less conservative sparseness constraint: To ensure that templates B are not aﬀected by noise we initially train the algorithm with a strong penalty term (large 2 β eﬀectively assuming strong noise power σN ). [sent-235, score-0.691]

91 Only spikes with large amplitudes remain after convergence and the templates are determined by only those strong spikes that have high SNR. [sent-236, score-1.225]

92 After extracting templates accurately, we retrain the model amplitudes A while keeping the templates B ﬁxed assuming now a weaker noise power (smaller β). [sent-237, score-1.564]

93 We have also derived a version of the model with observations X arranged as a matrix, as well as a version in which event timing is encoded explicitly as time delays τn following [8]. [sent-241, score-0.127]

94 This is particularly true for the semi-NMF algorithm since, provided a suﬃciently large K and without enforcing a sparsity constraint, the positivity constraint on A actually amounts to no constraint at all (identical templates with opposite sign can accomplish the same as allowing negative A). [sent-250, score-0.654]

95 For instance, without sparseness constraint on the amplitudes, a trivial solution in our examples above would be a template B1l with a single positive spike somewhere and another template B2l with a single negative spike, and all the time course encoded in An1 and An2 . [sent-251, score-0.702]

96 MISO identiﬁcation: The identiﬁability problem is compounded by the fact that the estimation of templates B in this present formulation represents a multiple-input singleoutput (MISO) system identiﬁcation problem. [sent-252, score-0.576]

97 Jordan, “Convex and semi-nonnegative matrix factorization for clustering and low-dimension representation,” Lawrence Berkeley National Laboratory, Tech. [sent-268, score-0.103]

98 Ding, “The relationships among various nonnegative matrix factorization methods for clustering,” in Proc. [sent-273, score-0.103]

99 Yin, “Enhancement of neural synchronization in the anteroventral cochlear nucleus. [sent-300, score-0.121]

100 Winter, “Spike waveforms in the anteroventral cochlear nucleus revisited,” in ARO midwinter meeting, no. [sent-311, score-0.191]

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