I Introduction
The extreme learning machine (ELM) [1] is an effective solution for Singlehiddenlayer feedforward networks (SLFNs) due to its unique characteristics, i.e., extremely fast learning speed, good generalization performance, and universal approximation capability [2]. Thus ELM has been widely applied in classification and regression [3].
The incremental ELM proposed in [2] achieves the universal approximation capability by adding hidden nodes one by one. However, it only updates the output weight for the newly added hidden node, and freezes the output weights of the existing hidden nodes. Accordingly those output weights are no longer the optimal leastsquares solution of the standard ELM algorithm. Then the inversefree algorithm was proposed in [4] to update the output weights of the added node and the existing nodes simultaneously, and the updated weights are identical to the optimal solution of the standard ELM algorithm. The ELM algorithm in [4] was based on an inversefree algorithm to compute the regularized pseudoinverse, which was deduced from an inversefree recursive algorithm to update the inverse of a Hermitian matrix.
Before the recursive algorithm to update the inverse was utilized in [4], it had been mentioned in previous literatures [5, 6, 7, 8, 9], while its improved version had been utilized in [9, 10]. Accordingly from the improved recursive algorithm [9, 10], we deduce a more efficient inversefree algorithm to update the regularized pseudoinverse, from which we develop the proposed ELM algorithm . Moreover, the proposed ELM algorithm computes the output weights directly from the updated inverse, to further reduce the computational complexity by avoiding the calculation of the regularized pseudoinverse. Lastly, instead of updating the inverse, the proposed ELM algorithm updates the factors of the inverse by the inverse factorization proposed in [11], since the recursive algorithm to update the inverse may introduce numerical instabilities in the processor units with the finite precision, which occurs only after a very large number of iterations [12].
This correspondence is organized as follows. Section ii@ describes the ELM model. Section iii@ introduces the existing inversefree ELM algorithm [4]. In Section iv@, we deduce the proposed inversefree ELM algorithms, and compare the expected computational complexities of the existing and proposed algorithms. Section v@ evaluates the existing and proposed algorithms by numerical experiments. Finally, we make conclusion in Section vi@.
Ii Architecture of the ELM
In the ELM model, the th input node, the th hidden node, and the th output node can be denoted as , , and , respectively, while all the input nodes, hidden nodes, and output nodes can be denoted as , , and , respectively. Accordingly the ELM model can be represented in a compact form as
(1) 
and
(2) 
where , ,
, and the activation function
is entrywise, i.e., for a matrix input . In (1), the activation function can be chosen as linear, sigmoid, Gaussian models, etc.Assume there are totally distinct training samples, and let and denote the th training input and the corresponding th training output, respectively, where . Then the input sequence and the output sequence in the training set can be represented as
(3) 
and
(4) 
respectively. We can substitute (3) into (1) to obtain
(5) 
where is the value sequence of all hidden nodes, and is the Kronecker product [4]. Then we can substitute (5) and (4) into (2) to obtain the actual training output sequence
(6) 
In an ELM, only the output weight is adjustable, while (i.e., the input weights) and (i.e., the biases of the hidden nodes) are randomly fixed. Denote the desired output as
. Then an ELM simply minimizes the estimation error
(7) 
by finding a leastsquares solution for the problem
(8) 
where denotes the Frobenius norm.
For the problem (8), the unique minimum norm leastsquares solution is [1]
(9) 
To avoid overfitting, the popular Tikhonov regularization [13, 14] can be utilized to modify (9) into
(10) 
where denotes the regularization factor. Obviously (9) is just the special case of (10) with . Thus in what follows, we only consider (10) for the ELM with Tikhonov regularization.
Iii The Existing InverseFree ELM Algorithm
In machine learning, it is a common strategy to increase the hidden node number gradually until the desired accuracy is achieved. However, when this strategy is applied in ELM directly, the matrix inverse operation in (
10) for the conventional ELM will be required when a few or only one extra hidden node is introduced, and accordingly the algorithm will be computational prohibitive. Accordingly an inversefree strategy was proposed in [4], to update the output weights incrementally with the increase of the hidden nodes. In each step, the output weights obtained by the inversefree algorithm are identical to the solution of the standard ELM algorithm using the inverse operation.Assume that in the ELM with hidden nodes, we add one extra hidden node, i.e., the hidden node
, which has the input weight row vector
and the bias . Then from (5) it can be seen that the extra row needs to be added to , i.e.,(11) 
where () denotes for the ELM with hidden nodes. In , , and what follows, we add the overline to emphasize the extra vector or scalar, which is added to the matrix or vector for the ELM with hidden nodes.
After is updated by (11), the conventional ELM updates the output weights by (10) that involves an inverse operation. To avoid that inverse operation, the algorithm in [4] utilizes an inversefree algorithm to update
(12) 
that is the regularized pseudoinverse of , and then substitutes (12) into (10) to compute the output weights by
(13) 
In [4], (i.e., for the ELM with hidden nodes) is computed from iteratively by
(14) 
where
(15) 
and , the column of , is computed by
(16) 
Iv Proposed InverseFree ELM Algorithms
Actually the inversefree recursive algorithm by (22) and (23c) had been mentioned in previous literatures [5, 6, 7, 8, 9], before it was deduced in [4] by utilizing the ShermanMorrison formula and the Schur complement. That inversefree recursive algorithm can be regarded as the application of the block matrix inverse lemma [5, p.30], and was called the lemma for inversion of blockpartitioned matrix [6, Ch. 14.12], [7, equation (16)]. To develop multipleinput multipleoutput (MIMO) detectors, the inversefree recursive algorithm was applied in [7, 8], and its improved version was utilized in [9, 10].




Existing Alg.  
Proposed Alg.  
Proposed Alg.  
Proposed Alg. 
Iva Derivation of Proposed ELM Algorithms
In the improved version [9, 10], equation (23c) has been simplified into [9, equation (20)]
(24a)  
(24b)  
(24c) 
Accordingly we can utilize (24c) to simplify (16) and (15) into
(25) 
and
(26) 
respectively, where can be computed by
(27) 
Moreover, from (25) and (26) we can deduce an efficient algorithm to update the output weight , i.e.,
(28) 
where
(29a)  
(29b) 
To further reduce the computational complexity, we can update the unique inverse by (21), (24c) and (22), and update the output weight by (28) where
(30a)  
(30b) 
are computed from and in . The derivation of (30b) is also in Appendix A.
Dataset+  Node  Weight Error  Output Error (training)  Output Error (testing)  Testing  

Kernel  Number  [4]  Alg. 1  Alg. 2  Alg. 3  [4]  Alg. 1  Alg. 2  Alg. 3  [4]  Alg. 1  Alg. 2  Alg. 3  MSE 
Airfoil  3  6e16  8e16  6e16  6e16  8e15  1e14  1e14  1e14  4e15  7e15  5e15  5e15  4.8e2 
+  100  2e11  3e11  1e8  2e11  5e12  8e12  4e9  5e12  3e12  5e12  2e9  2e12  1.1e2 
Gaussian  500  2e9  6e10  4e6  2e10  1e10  5e11  3e7  2e11  7e11  3e11  2e7  1e11  7.7e3 
Energy  3  2e14  1e14  1e14  7e15  7e14  5e14  4e14  4e14  3e14  2e14  2e14  2e14  3.0e2 
+  100  3e11  5e11  4e8  2e11  5e12  6e12  5e9  3e12  3e12  4e12  3e9  1e12  5.0e3 
Sigmoid  500  2e9  3e10  1e6  1e10  1e10  2e11  6e8  7e12  6e11  1e11  4e8  4e12  3.7e3 
Housing  3  3e16  4e16  7e16  5e16  2e15  3e15  5e15  4e15  1e15  2e15  3e15  2e15  8.6e2 
+  100  2e12  3e12  6e10  1e12  1e12  9e13  3e10  5e13  1e12  3e12  6e10  7e13  7.3e3 
Sine  500  4e10  6e11  4e8  2e11  5e11  7e12  6e9  3e12  4e10  7e11  4e8  3e11  5.4e3 
Protein  3  2e15  3e15  8e16  9e16  5e14  6e14  3e14  3e14  2e14  3e14  1e14  1e14  1.8e1 
+  100  2e11  2e11  2e9  3e11  4e11  5e11  4e9  6e11  2e11  2e11  2e9  3e11  5.6e2 
Triangular  500  2e9  1e9  3e6  1e9  1e9  1e9  2e6  1e9  9e10  7e10  1e6  6e10  4.9e2 
Since the processor units are limited in precision, the recursive algorithm utilized to update may introduce numerical instabilities, which occurs only after a very large number of iterations [12]. Thus instead of the inverse of (i.e., ), we can also update the inverse factors [11] of , since usually the factorization is numerically stable [15]. The inverse factors include the uppertriangular and the diagonal , which satisfy
(31) 
From (31) we can deduce
(32) 
where the lowertriangular is the conventional factor [15] of .
The inverse factors can be computed from directly by the inverse factorization in [11], i.e.,
(33a)  
(33b) 
where
(34a)  
(34b) 
IvB Summary and Complexity Analysis of ELM Algorithms
Firstly let us summarize the existing and proposed inversefree ELM algorithms, which all compute the output by (6), and compute the estimation error by (7). In (6) and (7), the output weight is required.
The existing inversefree ELM Algorithm [4] uses (15), (16) and (14) to update the regularized pseudoinverse , from which the output weight is computed by (13). The proposed Algorithm uses (21), (27), (25), (26) and (14) to update the regularized pseudoinverse , from which the output weight is computed by (29b) and (28). The proposed Algorithm uses (21), (24c) and (22) to update the unique inverse , from which the output weight is computed by (30b) and (28). The proposed Algorithm uses (21), (34b) and (33b) to update the factors of , from which the output weight is computed by (30b), (36) and (28).
Dataset+  Nodes  Speedups  

Kernel  Number  Alg. 1  Alg. 2  Alg. 3 
Airfoil+  100  2.43  7.99  5.66 
Gaussian  500  2.61  3.96  2.54 
Energy+  100  2.30  4.47  3.47 
Sigmoid  500  2.51  2.32  1.55 
Housing+  100  2.73  4.64  3.32 
Sine  500  2.77  1.92  1.41 
Protein+  100  2.54  19.04  16.28 
Triangular  500  2.66  22.09  19.29 
In the remainder of this subsection, we compare the expected flops (floatingpoint operations) of the existing ELM algorithm in [4] and the proposed ELM algorithms. Obviously flops are required to multiply a matrix by a matrix, and flops are required to sum two matrices in size [4].
In Table i@, we compare the flops of the existing ELM algorithm [4] and the proposed ELM algorithms , and . As in [4], the flops of the existing ELM algorithm do not include the entries for simplicity, since usually the ELM has large (the number of training examples) and (the number of hidden nodes). The flops of the proposed ELM algorithms do not include the entries that are or . Since usually , it can easily be seen from Table i@ that with respect to the existing ELM algorithm, the proposed ELM algorithms , and only require about , and of flops, respectively.
Dataset  Kernel  Mean/Variance 
Training  Testing  

ACC  SN  PE  MCC  ACC  SN  PE  MCC  
MAGIC  Gaussian  Mean  0.8645  0.9472  0.8584  0.6975  0.8618  0.9459  0.8561  0.6914 
Variance  0.0019  0.0018  0.0018  0.0045  0.0068  0.0058  0.0064  0.0153  
Sigmoid  Mean  0.8602  0.9468  0.8536  0.6877  0.8588  0.9458  0.8525  0.6844  
Variance  0.0019  0.0019  0.0018  0.0044  0.0065  0.0049  0.0063  0.0146  
Hardlim  Mean  0.8312  0.9277  0.8315  0.6202  0.8270  0.9249  0.8284  0.6104  
Variance  0.0038  0.0046  0.0045  0.0088  0.0069  0.0066  0.0083  0.0147  
Triangular  Mean  0.8592  0.9419  0.8555  0.6852  0.8561  0.9398  0.8532  0.6780  
Variance  0.0023  0.0025  0.0024  0.0052  0.0060  0.0051  0.0066  0.0131  
Sine  Mean  0.8640  0.9487  0.8569  0.6966  0.8620  0.9475  0.8552  0.6919  
Variance  0.0017  0.0016  0.0016  0.0040  0.0068  0.0058  0.0061  0.0152  
Musk  Gaussian  Mean  0.9453  0.6791  0.9522  0.7767  0.9412  0.6613  0.9396  0.7586 
Variance  0.0031  0.0193  0.0097  0.0135  0.0070  0.0321  0.0196  0.0238  
Sigmoid  Mean  0.9474  0.6925  0.9539  0.7862  0.9432  0.6745  0.9412  0.7679  
Variance  0.0030  0.0181  0.0097  0.0128  0.0068  0.0308  0.0189  0.0231  
Hardlim  Mean  0.9351  0.6185  0.9397  0.7309  0.9299  0.5969  0.9214  0.7075  
Variance  0.0036  0.0216  0.0128  0.0161  0.0076  0.0341  0.0247  0.0268  
Triangular  Mean  0.9447  0.6751  0.9528  0.7744  0.9406  0.6579  0.9390  0.7561  
Variance  0.0032  0.0191  0.0099  0.0137  0.0069  0.0318  0.0196  0.0232  
Sine  Mean  0.9462  0.6889  0.9479  0.7808  0.9419  0.6722  0.9326  0.7620  
Variance  0.0025  0.0145  0.0088  0.0105  0.0067  0.0301  0.0173  0.0218  
Adult  Gaussian  Mean  0.8362  0.9321  0.8612  0.5309  0.8359  0.9307  0.8626  0.5259 
Variance  0.0010  0.0018  0.0015  0.0034  0.0012  0.0020  0.0016  0.0041  
Sigmoid  Mean  0.8316  0.9313  0.8569  0.5160  0.8311  0.9297  0.8582  0.5101  
Variance  0.0014  0.0026  0.0023  0.0051  0.0017  0.0027  0.0023  0.0060  
Hardlim  Mean  0.8208  0.9314  0.8457  0.4786  0.8200  0.9298  0.8466  0.4711  
Variance  0.0023  0.0038  0.0034  0.0085  0.0026  0.0039  0.0034  0.0094  
Triangular  Mean  0.8367  0.9338  0.8607  0.5318  0.8366  0.9327  0.8620  0.5270  
Variance  0.0009  0.0018  0.0015  0.0032  0.0012  0.0019  0.0015  0.0040  
Sine  Mean  0.8377  0.9340  0.8616  0.5349  0.8377  0.9330  0.8630  0.5307  
Variance  0.0008  0.0016  0.0014  0.0028  0.0011  0.0017  0.0014  0.0035  
Diabetes  Gaussian  Mean  0.7973  0.6010  0.7572  0.5339  0.7681  0.5604  0.7048  0.4663 
Variance  0.0103  0.0251  0.0199  0.0239  0.0308  0.0668  0.0697  0.0684  
Sigmoid  Mean  0.7889  0.5746  0.7504  0.5124  0.7738  0.5548  0.7233  0.4781  
Variance  0.0091  0.0209  0.0166  0.0207  0.0312  0.0655  0.0703  0.0693  
Hardlim  Mean  0.7673  0.5380  0.7124  0.4602  0.7340  0.4892  0.6515  0.3819  
Variance  0.0159  0.0529  0.0278  0.0402  0.0348  0.0811  0.0775  0.0800  
Triangular  Mean  0.7964  0.5994  0.7558  0.5317  0.7674  0.5579  0.7046  0.4645  
Variance  0.0103  0.0249  0.0193  0.0238  0.0313  0.0677  0.0709  0.0704  
Sine  Mean  0.7972  0.5912  0.7633  0.5327  0.7721  0.5560  0.7174  0.4742  
Variance  0.0096  0.0228  0.0184  0.0220  0.0306  0.0662  0.0690  0.0679 
Notice that in the proposed ELM algorithm , computed in (27) can be utilized in (26) and (29a). The dominant computational load of the proposed ELM algorithm comes from (21), (27), (25) and (29b), of which the flops are , , and , respectively. Moreover, in the proposed ELM algorithms and , the dominant computational load comes from (21) and (30b), of which the flops are and , respectively.
V Numerical Experiments
We follow the simulations in [4], to compare the existing inversefree ELM algorithm and the proposed inversefree ELM algorithms on MATLAB software platform under a MicrosoftWindows Server with GB of RAM. We utilize a fivefold cross validation to partition the datasets into training and testing sets. To measure the performance, we employ the mean squared error (MSE) for regression problems, and employ four commonly used indices for classification problems, i.e., the prediction accuracy (ACC), the sensitivity (SN), the precision (PE) and the Matthews correlation coefficient (MCC). Moreover, the regularization factor is set to to avoid overfitting.
For the regression problem, we consider energy efficiency dataset [16], housing dataset [17], airfoil selfnoise dataset [18], and physicochemical properties of protein dataset [19]. For those datasets, different activation functions are chosen, which include Gaussian, sigmoid, sine and triangular. As Table iv@ in [4], Table ii@ shows the regression performance. In table ii@, the weight error and the output error are defined as and , respectively, where and are computed by an inversefree ELM algorithm, and and are computed by the standard ELM algorithm. We set the initial hidden node number to , and utilize the existing and proposed inversefree ELM algorithms to add the hidden nodes one by one till the hidden node number reaches . Table ii@ includes the simulation results for the hidden node numbers , and .
As observed from Table ii@, after iteration (i.e., the node number ), the weight error and the output error are less than . For the existing inversefree ELM algorithm and the proposed algorithms and , the weight error and the output error are less than after iterations (i.e., the node number ), and are not greater than after iterations (i.e., the node number ). However, for the proposed algorithms , the weight error and the output error are not greater than after iterations, and are not greater than after iterations, since the recursive algorithm to update introduces numerical instabilities after a very large number of iterations [12]. Overall, the standard ELM, the existing inversefree ELM algorithm and the proposed ELM algorithms , and achieve the same testing MSEs, which have been listed in the last column of Table ii@.
Algorithm  Gaussian  Sigmoid  Hardlim  Triangular  Sine  

Accuracy  Speedups  Accuracy  Speedups  Accuracy  Speedups  Accuracy  Speedups  Accuracy  Speedups  
Existing Alg.  
Proposed Alg. 1  3.41  3.77  
Proposed Alg. 2  45.50  44.04  
Proposed Alg. 3  26.28  31.04 
The speedups in training time of the proposed ELM algorithms , and over the existing inversefree ELM algorithm are shown in Table iii@, where we add just one node to reach and nodes, respectively, and we do simulations to compute the average training time. The speedups are computed by , i.e., the ratio between the training time of the existing ELM algorithm and that of the proposed ELM algorithm. As observed from Table iii@, all the proposed algorithms significantly accelerate the existing inversefree ELM algorithm.
For the classification problem, we consider MAGIC Gamma telescope dataset [20], musk dataset [21], adult dataset [22] and diabetes dataset [19]. For each dataset, five activation functions are simulated, i.e., Gaussian, sigmoid, Hardlim, triangular and sine. In the simulations, the standard ELM, the existing inversefree ELM algorithm and the proposed ELM algorithms , and achieve the same performance, which have been listed in Table iv@.
Lastly, in Table v@ we simulate the existing and proposed algorithms on the Modified National Institute of Standards and Technology (MNIST) dataset [23] with training images and testing images, to show the performance on big data. To give the testing accuracy, we set the initial hidden node number to , and utilize the existing and proposed ELM algorithms to add hidden nodes one by one till the hidden node number reaches . To give the speedups of the proposed algorithms over the existing algorithm, we compare the training time to reach nodes by adding one node, and do simulations to compute the average training time.
As observed from Table v@, the existing and proposed inversefree ELM algorithms bear the same testing accuracy, while all the proposed algorithms significantly accelerate the existing inversefree ELM algorithm. Moreover, from Table v@ and Table iii@, it can be seen that usually the proposed algorithm is faster than the proposed algorithm , and the proposed algorithm is faster than the proposed algorithm .
Vi Conclusion
To reduce the computational complexity of the existing inversefree ELM algorithm [4], in this correspondence we utilize the improved recursive algorithm [9, 10] to deduce the proposed ELM algorithms , and . The proposed algorithm includes a more efficient inversefree algorithm to update the regularized pseudoinverse . To further reduce the computational complexity, the proposed algorithm computes the output weights directly from the updated inverse , and avoids computing the regularized pseudoinverse . Lastly, instead of updating the inverse , the proposed ELM algorithm updates the factors of the inverse by the inverse factorization [11], since the inversefree recursive algorithm to update the inverse introduces numerical instabilities after a very large number of iterations [12]. With respect to the existing ELM algorithm, the proposed ELM algorithms , and are expected to require only , and of flops, respectively. In the numerical experiments, the standard ELM, the existing inversefree ELM algorithm and the proposed ELM algorithms 1, 2 and 3 achieve the same performance in regression and classification, while all the proposed algorithms significantly accelerate the existing inversefree ELM algorithm. Moreover, in the simulations, usually the proposed algorithm is faster than the proposed algorithm , and the proposed algorithm is faster than the proposed algorithm .
Appendix A Derivation of (25), (26), (27), (28), (29b) and (30b)
Substitute (11) and (18) into (12) to obtain
(37) 
Substitute (24c) into (22), which is then substituted into (37) to obtain , i.e.,
(38) 
To deduce (25), denote the second entry in the right side of (38) as
(39) 
into which substitute (24b) to obtain
(40) 
To deduce (26), substitute (24b) into the first entry in the right side of (38), and denote it as , i.e.,
(41) 
Then substitute (39) into (41) to obtain
(42) 
into which substitute (21) to obtain
(43) 
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