S EQUENTIAL D ENSITY R ATIO E STIMATION FOR S IMULTANEOUS O PTIMIZATION OF S PEED AND A CCURACY
Anonymous authors
Paper under double-blind review
A BSTRACT
Classifying sequential data as early and as accurately as possible is a challenging yet critical problem, especially when a sampling cost is high.
One algorithm that achieves this goal is the sequential probability ratio test (SPRT), which is known as Bayes-optimal: it can keep the expected number of data samples as small as possible, given the desired error upper-bound.
However, the original
SPRT makes two critical assumptions that limit its application in real-world scenarios: (i) samples are independently and identically distributed, and (ii) the likelihood of the data being derived from each class can be calculated precisely.
Here, we propose the SPRT-TANDEM, a deep neural network-based SPRT algorithm that overcomes the above two obstacles.
The SPRT-TANDEM sequentially estimates the log-likelihood ratio of two alternative hypotheses by leveraging a novel Loss function for Log-Likelihood Ratio estimation (LLLR) while allowing correlations up to N ( 2 N ) preceding samples.
In tests on one original and two public video databases, Nosaic MNIST, UCF101, and SiW, the SPRT-TANDEM achieves statistically signiﬁcantly better classiﬁcation accuracy than other baseline classiﬁers, with a smaller number of data samples.
The code and Nosaic MNIST are publicly available at https://anonymous.4open.science/ r/8e802b42-ec6f-4545-b34e-fb320cba4c4d/#home .
I NTRODUCTION
The sequential probability ratio test, or SPRT, was originally invented by Abraham Wald, and an equivalent approach was also independently developed and used by Alan Turing in the 1940s (Good, 1979; Simpson, 2010; Wald, 1945).
SPRT calculates the log-likelihood ratio (LLR) of two competing hypotheses and updates the LLR every time a new sample is acquired until the LLR reaches one of the two thresholds for alternative hypotheses (Figure 1).
Wald and his colleagues proved that when sequential data are sampled from independently and identically distributed (i.i.d.) data, SPRT can minimize the required number of samples to achieve the desired upper-bounds of false positive and false negative rates comparably to the Neyman-Pearson test, known as the most powerful likelihood test (Wald & Wolfowitz, 1948) (see also Theorem (A.5) in Appendix A).
Note that Wald used the i.i.d.
assumption only for ensuring a ﬁnite decision time (i.e., LLR reaches a threshold within ﬁnite steps) and for facilitating LLR calculation: the non-i.i.d.
property does not affect other aspects of the SPRT including the error upper bounds (Wald, 1947).
More recently, Tartakovsky et al.
veriﬁed that the non-i.i.d.
SPRT is optimal or at least asymptotically optimal as the sample size increases (Tartakovsky et al., 2014), opening the possibility of potential applications of the SPRT to non-i.i.d.
data series.
About 70 years after Wald’s invention, neuroscientists found that neurons in the part of the primate brain called the lateral intraparietal cortex (LIP) showed neural activities reminiscent of the SPRT (Kira et al., 2015); when a monkey sequentially collects random pieces of evidence to make a binary choice, LIP neurons show activities proportional to the LLR.
Importantly, the time of the decision can be predicted from when the neural activity reaches a ﬁxed threshold, the same as the
SPRT’s decision rule.
Thus, the SPRT, the optimal sequential decision strategy, was re-discovered to be an algorithm explaining primate brains’ computing strategy.
It remains an open question, however, what algorithm will be used in the brain when the sequential evidence is correlated, non-i.i.d. series.
[Figure]
The SPRT is now used for several engineering applications (Cabri et al., 2018;
Chen et al., 2017;
Kulldorff et al., 2011).
However, its i.i.d.
assumption is too crude for it to be applied to other real-world scenarios, including time-series classiﬁcation, where data are highly correlated, and key dynamic features for classiﬁcation often extend across more than one data point, violating the i.i.d. assumption.
Moreover, the LLR of alternative hypotheses needs to be calculated as precisely as possible, which is infeasible in many practical applications.
In this paper, we overcome the above difﬁculties by using an SPRT-based algorithm that Treats data series As an N-th orDEr Markov process (SPRT-TANDEM), aided by a sequential probability density ratio estimation based on deep neural networks.
A novel Loss function for Log-Likelihood Ratio estimation (LLLR) efﬁciently estimates the density ratio that let the SPRT-TANDEM approach Bayes-optimality; in other words, LLLR optimizes classiﬁcation speed and accuracy at the same time.
The SPRT-TANDEM can classify non-i.i.d.
data series with user-deﬁned model complexity by changing N ( 2 N ) , the order of approximation, to deﬁne the number of past samples on which the given sample depends.
By dynamically changing the number of samples used for classiﬁcation, the
SPRT-TANDEM can maintain high classiﬁcation accuracy while minimizing the sample size as much as possible.
Moreover, the SPRT-TANDEM enables a user to ﬂexibly control the speed-accuracy tradeoff without additional training, making it applicable to various practical applications.
We test the SPRT-TANDEM on our new database, Nosaic MNIST (NMNIST), in addition to the publicly available UCF101 action recognition database (Soomro et al., 2012) and Spooﬁng in the
Wild (SiW) database (Liu et al., 2018).
Two-way analysis of variance (ANOVA, Fisher (1925)) followed by a Tukey-Kramer multi-comparison test (Tukey, 1949; Kramer, 1956) shows that our proposed SPRT-TANDEM provides statistically signiﬁcantly higher accuracy than other ﬁxed-length and variable-length classiﬁers at a smaller number of data samples, making Wald’s SPRT applicable even to non-i.i.d.
data series.
Our contribution is ﬁvefold:
We invented a deep neural network-based algorithm, SPRT-TANDEM, which enables Wald’s SPRT on arbitrary sequential data without knowing the true LLR. The SPRT-TANDEM extends the SPRT to non-i.i
d. data series without knowing the true LLR.
With a novel loss, LLLR, the SPRT-TANDEM sequentially estimates LLR to optimize speed and accuracy simultaneously.
The SPRT-TANDEM can control the speed-accuracy tradeoff without additional training.
We introduce Nosaic MNIST, a novel early-classiﬁcation database.
R ELATED WORK
The SPRT-TANDEM has multiple interdisciplinary intersections with other ﬁelds of research: Wald’s classical SPRT, probability density estimation, neurophysiological decision making, and time-series classiﬁcation.
The comprehensive review is left to Appendix B, while in the following, we introduce the SPRT, probability density estimation algorithms, and early classiﬁcation of the time series.
Sequential Probability Ratio Test (SPRT).
The SPRT, denoted by  ⇤ , is deﬁned as the tuple of a decision rule and a stopping rule (Tartakovsky et al., 2014; Wald, 1947):
Deﬁnition 2.1.
Sequential Probability Ratio Test (SPRT).
Let  t as the LLR at time t .
Given the absolute values of lower and upper decision threshold, a 0  0 and a 1  0 , SPRT,  ⇤ , is deﬁned as
[Equation]
where the decision rule d ⇤ and stopping time ⌧ ⇤ are
[Equation]
We review the proof of optimality in Appendix A.4, while Figure 1 shows an intuitive explanation.
Probability density ratio estimation.
Instead of estimating numerator and denominator of a density ratio separately, the probability density ratio estimation algorithms estimate the ratio as a whole, reducing the degree of freedom for more precise estimation (Sugiyama et al., 2010; 2012).
Two of the probability density ratio estimation algorithms that closely related to our work are the probabilistic classiﬁcation (Bickel et al., 2007; Cheng & Chu, 2004; Qin, 1998) and ratio matching (Kanamori et al., 2009; Sugiyama et al., 2008; Tsuboi et al., 2009) algorithms.
As we show in Section 3 and 4, the SPRT-TANDEM sequentially estimates the LLR by combining the two algorithms.
Early classiﬁcation of time series.
To make decision time as short as possible, algorithms for early classiﬁcation of time series can handle variable length of data (Mori et al., 2018;
Mori et al., 2016;
Xing et al., 2009; 2012) to minimize high sampling costs (e.g., medical diagnostics (Evans et al., 2015;
Grifﬁn & Moorman, 2001), or stock crisis identiﬁcation (Ghalwash et al., 2014)).
Leveraging deep neural networks is no exception in the early classiﬁcation of time series (Dennis et al., 2018; Suzuki et al., 2018).
Long short-term memory (LSTM)-s/LSTM-m impose monotonicity on classiﬁcation score and inter-class margin, respectively, to speed up action detection (Ma et al., 2016).
Early and
Adaptive Recurrent Label ESTimator (EARLIEST) combines reinforcement learning and a recurrent neural network to decide when to classify and assign a class label (Hartvigsen et al., 2019).
P ROPOSED ALGORITHM : SPRT-TANDEM
In this section, we propose the TANDEM formula, which provides the N -th order approximation of the LLR with respect to posterior probabilities.
The i.i.d. assumption of Wald’s SPRT greatly simpliﬁes the LLR calculation at the expense of the precise temporal relationship between data samples.
On the other hand, incorporating a long correlation among multiple data may improve the LLR estimation; however, calculating too long a correlation may potentially be detrimental in the following cases.
First, if a class signature is signiﬁcantly shorter than the correlation length in consideration, uninformative data samples are included in calculating LLR, resulting in a late or wrong decision (Campos et al., 2018).
Second, long correlations require calculating a long-range of backpropagation, prone to vanishing gradient problem (Hochreiter et al., 2001).
Thus, we relax the i.i.d.
assumption by keeping only up to the N -th order correlation to calculate the LLR.
The TANDEM formula.
Here, we introduce the TANDEM formula, which computes the approximated LLR, the decision value of the SPRT-TANDEM algorithm.
The data series is approximated as an N -th order Markov process.
For the complete derivation of the 0th (i.i.d.), 1st, and N -th order
TANDEM formula, see Appendix C. Given a maximum timestamp T 2 N , let X (1 ,T ) and y be a sequential data X (1 ,T ) :
= { x ( t ) }
Tt =1 and a class label y 2 { 1 , 0 } , respectively, where x ( t ) 2 R d x and d x 2 N .
By using Bayes’ rule with the N -th order Markov assumption, the joint LLR of data at a timestamp t is written as follows:
[Equation]
[Figure]
(see Equation (82) and (83) in Appendix C for the full formula).
Hereafter we use terms k -let or multiplet to indicate the posterior probabilities, p ( y | x (1) , ..., x ( k ) )
=
p ( y | X (1 ,k ) ) that consider correlation across k data points.
The ﬁrst two terms of the TANDEM formula (Equation (4)), N + 1let and N -let, have the opposite signs working in “tandem” adjusting each other to compute the LLR.
The third term is a prior (bias) term.
In the experiment, we assume a ﬂat prior or zero bias term, but a user may impose a non-ﬂat prior to handling the biased distribution of a dataset.
The TANDEM formula can be interpreted as a realization of the probability matching approach of the probability density estimation, under an N -th order Markov assumption of data series.
Neural network that calculates the SPRT-TANDEM formula.
The SPRT-TANDEM is designed to explicitly calculate the N -th order TANDEM formula to realize sequential density ratio estimation, which is the critical difference between our SPRT-TANDEM network and other architecture based on convolutional neural networks (CNNs) and recurrent neural networks (RNN).
Figure 2 illustrates a conceptual diagram explaining a generalized neural network structure, in accordance with the 1st-order TANDEM formula for simplicity.
The network consists of a feature extractor and a temporal integrator (highlighted by red and blue boxes, respectively).
They are arbitrary networks that a user can choose depending on classiﬁcation problems or available computational resources.
The feature extractor and temporal integrator are separately trained because we ﬁnd that this achieves better performance than the end-to-end approach (also see Appendix D).
The feature extractor outputs single-frame features (e.g., outputs from a global average pooling layer), which are the input vectors of the temporal integrator.
The output vectors from the temporal integrator are transformed with a fully-connected layer into two-dimensional logits, which are then input to the softmax layer to obtain posterior probabilities.
They are used to compute the LLR to run the SPRT (Equation (2)).
Note that during the training phase of the feature extractor, the global average pooling layer is followed by a fully-connected layer for binary classiﬁcation.
How to choose the hyperparameter N ?
By tuning the hyperparameter N , a user can efﬁciently boost the model performance depending on databases; in Section 5, we change N to visualize the model performance as a function of N .
Here, we provide two ways to choose N .
One is to choose N based on the speciﬁc time scale, a concept introduced in Appendix D, where we describe in detail how to guess on the best N depending on databases.
The other is to use a hyperparameter tuning algorithm such as Optuna (Akiba et al., 2019) to choose N objectively.
Note that tuning N is not computationally expensive, because N is only related to the temporal integrator, not the feature extractor.
In fact, the temporal integrator’s training speed is much faster than that of the feature extractor: 9 mins/epoch vs. 10 hrs/epoch ( N = 49 , NVIDIA RTX2080Ti, SiW database).
LLLR AND MULTIPLET CROSS - ENTROPY LOSS
Given a maximum timestamp T 2 N and dataset size M 2 N , let S :
= { ( X (1 ,T ) i , y i ) } Mi =1 be a sequential dataset.
Training our network to calculate the TANDEM formula involves the following loss functions in combination: (i) the Loss for Log Likelihood Ratio estimation (LLLR), L LLR , and (ii) multiplet cross-entropy loss, L multiplet .
The total loss, L total is deﬁned as
[Equation] Loss for Log-Likelihood Ratio estimation (LLLR). The SPRT is Bayes-optimal as long as the true LLR is available; however, the true LLR is often inaccessible under real-world scenarios. To empirically estimate the LLR with the TANDEM formula, we propose the LLLR, which minimizes the Kullback-Leibler divergence between the estimated and the true LLRs:
[Equation]
where  is the sigmoid function. The LLLR is interpreted as a ratio matching approach of the probability density ratio estimation; in fact, as shown in Appendix E.3, the LLLR is a normalized variant of KLIEP (Khan et al., 2019; Sugiyama et al., 2008), a ratio matching algorithm minimizing the Kullback-Leibler divergence between the estimated and true density ratio. The LLLR normalizes the objective function of KLIEP, because it is unbounded in the original formulation and often causes instability of training. In contrast, the LLLR can readily be used together with conventional loss functions such as the cross-entropy loss.
Here we emphasize the contributions of the LLLR again: the LLLR enables us to conduct the stable LLR estimation and, therefore, to perform the SPRT, which provably provides the optimal accuracy and stopping time. On the other hand, these two objectives are achieved with separate loss sub-functions in previous works (Mori et al., 2018; Hartvigsen et al., 2020); however, only the LLLR sufﬁces in our model.
Multiplet cross-entropy loss.
Though the LLLR is sufﬁcient to estimate LLR, we add binary cross-entropy losses to further facilitate the training. We call them multiplet cross-entropy loss here, and deﬁned as:
[Equation]
where
[Equation]
We use ˆ p to highlight a probability density estimated by a neural network.
Minimizing the multiplet cross-entropy loss is equivalent to minimizing the Kullback-Leibler divergence (Kullback & Leibler, 1951) of the estimated posterior k -let
ˆ p ( y i | x
( t  k +1)
i , ..., x ( t ) i ) and the true posterior p ( y i | x ( t  k +1) i , ..., x ( t ) i ) (shown in Appendix F), which is a consistent objective with the LLLR and thus the multiplet loss accelerates the training.
Note also that the multiplet loss optimizes all the logits output from the temporal integrator, unlike the LLLR.
Appendix E.4 empirically shows that the multiplet loss statistically signiﬁcantly boosts the performance of the SPRT-TANDEM.
E XPERIMENTS AND RESULTS
In the following experiments, we use two quantities as evaluation criteria:
(i) balanced accuracy, the arithmetic mean of the true positive and true negative rates, and (ii) mean hitting time, the average number of data samples used for classiﬁcation.
Note that the balanced accuracy is robust to class imbalance (Luque et al., 2019), and is equal to accuracy on balanced datasets.
Evaluated public databases are NMNIST, UCF, and SiW. Training, validation, and test datasets are split and ﬁxed throughout the experiment.
We selected three early-classiﬁcation models (LSTM-s
[Figure]
(Ma et al., 2016), LSTM-m (Ma et al., 2016), and EARLIEST (Hartvigsen et al., 2019)) and one ﬁxed-length classiﬁer (3DResNet (Hara et al., 2017)), as baseline models.
All the early-classiﬁcation models share the same feature extractor as that of the SPRT-TANDEM for a fair comparison.
Hyperparameters of all the models are optimized with Optuna unless otherwise noted so that no models are disadvantaged by choice of hyperparameters.
See Appendix G for the search spaces and ﬁxed ﬁnal parameters.
After ﬁxing hyperparameters, experiments are repeated with different random seeds to obtain statistics.
In each of the training runs, we evaluate the validation set after each training epoch and then save the weight parameters if the balanced accuracy on the validation set updates the largest value.
The last saved weights are used as the model of that run.
The model evaluation is performed on the test dataset.
During the test stage of the SPRT-TANDEM, we used various values of the SPRT thresholds to obtain a range of balanced accuracy-mean hitting time combinations to plot a speed-accuracy tradeoff (SAT) curve.
If all the samples in a video are used up, the thresholds are collapsed to a 1 = a 0 = 0 to force a decision.
To objectively compare all the models with various trial numbers, we conducted the two-way ANOVA followed by the Tukey-Kramer multi-comparison test to compute statistical signiﬁcance.
For the details of the statistical test, see Appendix H.
We show our experimental results below.
Due to space limitations, we can only show representative results.
For more details, see Appendix I.
For our computing infrastructure, see Appendix J.
Nosaic MNIST (Noise + mosaic MNIST) database.
We introduce a novel dataset, NMNIST, whose video is buried with noise at the ﬁrst frame, and gradually denoised toward the last, 20th frame (see Appendix K for example data).
The motivation to create NMNIST instead of using a preexisting time-series database is as follows: for simple video databases such as Moving MNIST (MMNIST, (Srivastava et al., 2015)), each data sample contains too much information so that well-trained classiﬁers can correctly classify a video only with one or two frames (see Appendix L for the results of the SPRT-TANDEM and LSTM-m on MMNIST).
We design a parity classiﬁcation task, classifying
9 digits into an odd or even class.
The training, validation, and test datasets contain 50,000, 10,000, and 10,000 videos with frames of size 28 ⇥ 28 ⇥(gray scale).
Each pixel value is divided by 127 . 5 , before subtracted by 1 .
The feature extractor of the
SPRT-TANDEM is ResNet-
(He et al., 2016a), with the ﬁnal output reduced to 128 channels.
The temporal integrator is a peephole-LSTM (Gers & Schmidhuber, 2000; Hochreiter & Schmidhuber, 1997), with hidden layers of 128 units.
The total numbers of trainable parameters on the feature extractor and temporal integrator are 6.9M and 0.1M, respectively.
We train 0th, 1st, 2nd, 3rd, 4th, 5th, 10th, and 19th order SPRT-TANDEM networks.
LSTM-s / LSTM-m and EARLIEST use peephole-LSTM and LSTM, respectively, both with hidden layers of 128 units.
3DResNet has layers with 128 ﬁnal output channels so that the total number of trainable parameters is in the same order (7.7M) as that of the SPRT-TANDEM.
Figure 3a and Table 1 show representative results of the experiment.
Figure 3d shows example
LLR trajectories calculated with the 10th order SPRT-TANDEM.
The SPRT-TANDEM outperforms other baseline algorithms by large margins at all mean hitting times.
The best performing model is the 10th order TANDEM, which achieves statistically signiﬁcantly higher balanced accuracy than other algorithms ( p -value < 0 . 001 ).
The superiority of the SPRT-TANDEM over other algorithms indicates that the SPRT-TANDEM sequentially computes and minimizes the distance between the estimated LLR and the true LLR, approaching Bayes optimally if not reaches.
[Table]
UCF101 action recognition database.
To create a more challenging task, we selected two classes, handstand-pushups and handstand-walking, from the 101 classes in the UCF database.
At a glimpse of one frame, the two classes are hard to distinguish.
Thus, to correctly classify these classes, temporal information must be properly used.
We resize each video’s duration as multiples offrames and sample every 50 frames with 25 frames of stride as one data.
Training, validation, and test datasets contain 1026, 106, and 105 videos with frames of size 224 ⇥ 224 ⇥ 3 , randomly cropped to 200 ⇥ 200 ⇥ 3 at training.
The mean and variance of a frame are normalized to zero and one, respectively.
The feature extractor of the SPRT-TANDEM is ResNet-50 (He et al., 2016b), with the ﬁnal output reduced to 64 channels.
The temporal integrator is a peephole-LSTM, with hidden layers of 64 units.
The total numbers of trainable parameters in the feature extractor and temporal integrator are 26K and 33K, respectively.
We train 0th, 1st, 2nd, 3rd, 5th, 10th, 19th, 24th, and 49th-order
SPRT-TANDEM.
LSTM-s / LSTM-m and EARLIEST use peephole-LSTM and LSTM, respectively, both with hidden layers of 64 units.
3DResNet has 50 layers with 64 ﬁnal output channels so that the total number of trainable parameters (52K) is on the same order as that of the SPRT-TANDEM.
Figure 3b shows representative results of the experiment.
The best performing model is the 10th order
TANDEM, which achieves statistically signiﬁcantly higher balanced accuracy than other models ( p -value < 0 . 001 ).
The superiority of the higher-order TANDEM indicates that a classiﬁer needs to integrate longer temporal information in order to distinguish the two classes (also see Appendix D).
[Table]
Spooﬁng in the Wild (SiW) database.
To test the SPRT-TANDEM in a more practical situation, we conducted experiments on the SiW database.
We use a sliding window of 50 frames-length andframes-stride to sample data, which yields training, validation
[Table]
and test datasets of 46,729, 4,
[Table]
and
43,878 videos of live or spooﬁng face.
Each frame is resized to 256 ⇥ 256 ⇥ 3 pixels and randomly cropped to 244 ⇥ 244 ⇥ 3 at training.
The mean and variance of a frame are normalized to zero and one, respectively.
The feature extractor of the SPRT-TANDEM is ResNet-152, with the ﬁnal output reduced to 512 channels.
The temporal integrator is a peephole-LSTM, with hidden layers of 512 units.
The total number of trainable parameters in the feature extractor and temporal integrator is 3.7M and 2.1M, respectively.
We train 0th, 1st, 2nd, 3rd, 5th, 10th, 19th, 24th, and 49th-order
SPRT-TANDEM networks.
LSTM-s / LSTM-m and EARLIEST use peephole-LSTM and LSTM, respectively, both with hidden layers of 512 units.
3DResNet has 101 layers with 512 ﬁnal output channels so that the total number of trainable parameters (5.3M) is in the same order as that of the
SPRT-TANDEM.
Optuna is not applied due to the large database and network size.
Figure 3c shows representative results of the experiment.
The best performing model is the 10th order
TANDEM, which achieves statistically signiﬁcantly higher balanced accuracy than other models ( p -value < 0 . 001 ).
The superiority of the lower-order TANDEM indicates that each video frame contains a high amount of information necessary for the classiﬁcation, imposing less need to collect a large number of frames (also see Appendix D).
[Table]
Ablation study.
To understand contributions of the L LLR and L multiplet to the SAT curve
[Table]
we conduct an ablation study.
The 1st-order SPRT-TANDEM is trained with L LLR only, L multiplet only
[Table]
and both L LLR and L multiplet .
The hyperparameters of the three models are independently optimized using Optuna (see Appendix G).
The evaluated database and model are NMNIST and the 1st-order
SPRT-TANDEM, respectively.
Figure 3e shows the three SAT curves.
The result shows that L LLR leads to higher classiﬁcation accuracy
[Table]
whereas L multiplet enables faster classiﬁcation.
The best performance is obtained by using both L LLR and L multiplet .
We also conﬁrmed this tendency with the 19th order SPRT-TANDEM, as shown in Appendix M.
SPRT vs. Neyman-Pearson test.
As we discuss in Appendix A, the Neyman-Person test is the optimal likelihood ratio test with a ﬁxed number of samples.
The SPRT, however, reaches the accuracy with much smaller data samples in the early classiﬁcation of time series,
i
[Table]
e., in the test with ﬂexible numbers of samples.
To experimentally test this prediction, we compare the SPRT-TANDEM and corresponding the Neyman-Pearson test.
By using the calculated LLR trajectory, the Neyman-Pearson test classiﬁes the entire data into two classes at each number of frames, using threshold  = 0 .
Results support the theoretical prediction, as shown in Figure 3f: the Neyman-Pearson test needs a larger number of samples than the SPRT-TANDEM at the early phase with few frames, asymptotically approaches the SPRT-TANDEM at the later phase with many frames.
C ONCLUSION
We presented the SPRT-TANDEM, a novel algorithm making Wald’s SPRT applicable to arbitrary data series without knowing the true LLR.
Leveraging deep neural networks and the novel loss function, LLLR, the SPRT-TANDEM minimizes the distance of the true LLR and the LLR sequentially estimated with the TANDEM formula, enabling simultaneous optimization of speed and accuracy.
Tested on the three publicly available databases, the SPRT-TANDEM achieves statistically significantly higher accuracy over other existing algorithms with a smaller number of data points.
The
SPRT-TANDEM enables a user to control the speed-accuracy tradeoff without additional training, opening up various potential applications where either high-accuracy or high-speed is required.