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1.449 -2.898 l
0 -2.174 l
-1.449 -2.898 l
0 0 l
s
Q
BT
/TT4 1 Tf
9.1079 0 0 9.1079 261.7235 382.1124 Tm
<005b>Tj
-12 11.8182 TD
<005c>Tj
ET
Q
BT
/F2 1 Tf
8.7176 0 0 8.7176 83.0026 328.6363 Tm
[(Figure)-249.9(2.)-250(P)15(arameters)-250(of)-250(Piece)25(wise)-250(Linear)-250(Unit)]TJ
9.6862 0 0 9.6862 60.345 303.7962 Tm
[(W)80(e)-313<93727374>-313(de\223ne)-313(the)-313(proposed)-313(acti)25(v)25(ation)-313(function:)-436(Piece-)]TJ
-1.2 -1.2001 TD
[(wise)-283(Linear)-283(Unit.)-408(As)-283(sho)25(wn)-283(in)-283(Figure)]TJ
/Cs6 cs .92941 .1098 .14118 scn
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(2)Tj
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[(,)-291(PWLU)-283(is)-283(de\223ned)]TJ
-16.1629 -1.2 TD
[(by)-250(follo)25(wing)-250(parameters:)]TJ
/F5 1 Tf
1 -1.9746 TD
<80>Tj
/F2 1 Tf
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[(Number)-250(of)-250(interv)25(als)]TJ
/F6 1 Tf
8.2789 0 TD
(N)Tj
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-9.2789 -1.9874 TD
<80>Tj
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[(Left)-250(boundary)]TJ
/F6 1 Tf
5.943 0 TD
(B)Tj
/F9 1 Tf
6.7803 0 0 6.7803 133.0062 240.7211 Tm
(L)Tj
/F2 1 Tf
9.6862 0 0 9.6862 138.8043 242.1731 Tm
[(,)-250(right)-250(boundary)]TJ
/F6 1 Tf
6.666 0 TD
(B)Tj
/F9 1 Tf
6.7803 0 0 6.7803 210.7204 240.7211 Tm
(R)Tj
/F5 1 Tf
9.6862 0 0 9.6862 58.4083 222.9241 Tm
<80>Tj
/F2 1 Tf
.9999 0 TD
[(y-axis)-250(v)25(alue)-250(of)]TJ
/F6 1 Tf
6.168 0 TD
(N)Tj
/F7 1 Tf
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(+1)Tj
/F2 1 Tf
1.75 0 TD
0 Tc
[(demarcation)-250(points)]TJ
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(Y)Tj
/F9 1 Tf
6.7803 0 0 6.7803 237.8085 221.4721 Tm
(P)Tj
/F5 1 Tf
9.6862 0 0 9.6862 58.4083 203.676 Tm
<80>Tj
/F2 1 Tf
.9999 0 TD
[(Left)-250(most)-250(slope)]TJ
/F6 1 Tf
6.472 0 TD
(K)Tj
/F9 1 Tf
6.7803 0 0 6.7803 139.0093 202.2221 Tm
(L)Tj
/F2 1 Tf
9.6862 0 0 9.6862 144.8083 203.676 Tm
[(,)-250(right)-250(most)-250(slope)]TJ
/F6 1 Tf
7.1949 0 TD
(K)Tj
/F9 1 Tf
6.7803 0 0 6.7803 222.7254 202.2221 Tm
(R)Tj
/F2 1 Tf
9.6862 0 0 9.6862 60.345 184.55 Tm
[(The)-335(number)-336(of)-335(interv)25(als)]TJ
/F6 1 Tf
10.2035 0 TD
(N)Tj
/F2 1 Tf
1.2479 0 TD
[(is)-335(a)-336(h)5(yperparameter)40(,)-356(which)]TJ
-12.6514 -1.2001 TD
[(in\224uences)-295(the)-294(\223tting)-294.9(ability)-294(of)-295(PWLU.)-294(A)-295(lar)18(ger)-294(number)-295(of)]TJ
0 -1.2 TD
[(interv)25(als)-347(brings)-346.9(higher)-347.1(de)15(grees)-347(of)-347(freedom,)-372(increasing)-347(the)]TJ
0 -1.2001 TD
[(capacity)-255(of)-255(the)-254(model.)-325(The)-254(left)-255(and)-255(right)-255(boundary)]TJ
/F6 1 Tf
20.4976 0 TD
(B)Tj
/F9 1 Tf
6.7803 0 0 6.7803 254.6125 148.226 Tm
(L)Tj
/F6 1 Tf
9.6862 0 0 9.6862 260.4115 149.679 Tm
.1678 Tc
(,B)Tj
/F9 1 Tf
6.7803 0 0 6.7803 272.0625 148.226 Tm
0 Tc
(R)Tj
/F2 1 Tf
9.6862 0 0 9.6862 48.7217 138.055 Tm
[(de\223nes)-376(the)-376(ef)25(fecti)25(v)15(e)-376(re)15(gion)-376(where)-377(PWLU)-376(mainly)-376(focuses)]TJ
0 -1.2 TD
(on.)Tj
/F7 1 Tf
1.5716 0 TD
([)Tj
/F6 1 Tf
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(B)Tj
/F9 1 Tf
6.7803 0 0 6.7803 73.9828 124.9789 Tm
(L)Tj
/F6 1 Tf
9.6862 0 0 9.6862 79.7805 126.4319 Tm
.1678 Tc
(,B)Tj
/F9 1 Tf
6.7803 0 0 6.7803 91.433 124.9789 Tm
0 Tc
(R)Tj
/F7 1 Tf
9.6862 0 0 9.6862 97.7867 126.4319 Tm
(])Tj
/F2 1 Tf
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(is)Tj
/F4 1 Tf
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(uniformly)Tj
/F2 1 Tf
4.1429 0 TD
[(di)25(vided)-254(into)]TJ
/F6 1 Tf
5.0386 0 TD
(N)Tj
/F2 1 Tf
1.1664 0 TD
[(interv)25.1(als,)-255(lea)19.9(ving)]TJ
/F6 1 Tf
-16.8659 -1.2 TD
(N)Tj
/F7 1 Tf
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.2838 Tc
(+1)Tj
/F2 1 Tf
1.8923 0 TD
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[(demarcation)-332(points.)-555(Each)-331(demarcation)-331.9(point)-332(has)-331(a)]TJ
22.8847 63.1398 TD
[(corresponding)-224.9(y-axis)-224.9(v)25(alue)]TJ
/F6 1 Tf
10.9807 0 TD
(Y)Tj
/F9 1 Tf
6.7803 0 0 6.7803 412.2778 724.941 Tm
(P)Tj
/F2 1 Tf
9.6862 0 0 9.6862 418.7368 726.394 Tm
[(,)-230(which)-225(determines)-225(the)-225(shape)]TJ
-12.228 -1.2 TD
[(of)-363(PWLU.)-364(Be)15(yond)]TJ
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/F6 1 Tf
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(B)Tj
/F9 1 Tf
6.7803 0 0 6.7803 388.8118 713.3181 Tm
(L)Tj
/F6 1 Tf
9.6862 0 0 9.6862 394.6098 714.7711 Tm
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(,B)Tj
/F9 1 Tf
6.7803 0 0 6.7803 406.2618 713.3181 Tm
0 Tc
(R)Tj
/F7 1 Tf
9.6862 0 0 9.6862 412.6158 714.7711 Tm
(])Tj
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[(,)-392(w)0(e)-363(use)-364(tw)10(o)-363(slopes)]TJ
/F6 1 Tf
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(K)Tj
/F9 1 Tf
6.7803 0 0 6.7803 506.684 713.3181 Tm
(L)Tj
/F2 1 Tf
9.6862 0 0 9.6862 516.002 714.7711 Tm
(and)Tj
/F6 1 Tf
-22.2696 -1.2001 TD
(K)Tj
/F9 1 Tf
6.7803 0 0 6.7803 308.5196 701.6951 Tm
(R)Tj
/F2 1 Tf
9.6862 0 0 9.6862 317.2956 703.1471 Tm
[(to)-250(control)-250(the)-250(re)15(gion)-250(out)-250(of)-250(the)-250(boundaries.)]TJ
-.5553 -1.2 TD
[(Gi)25(v)15(e)0(n)-370(these)-369(parameters,)-399(the)-370(forw)10(ard)-369(pass)-370(of)-369(PWLU)-370(is)]TJ
-1.2 -1.2 TD
[(sho)25(wn)-250(in)-250(Equation)]TJ
/Cs6 cs .92941 .1098 .14118 scn
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(1)Tj
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-2.1711 -2.0806 TD
(PWLU)Tj
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(N)Tj
/F7 1 Tf
9.6862 0 0 9.6862 393.7198 659.7479 Tm
(\()Tj
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[(x,)-167.3(B)]TJ
/F9 1 Tf
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9.6862 0 0 9.6862 420.4728 659.7479 Tm
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(,B)Tj
/F9 1 Tf
6.7803 0 0 6.7803 432.1248 658.2949 Tm
0 Tc
(R)Tj
/F6 1 Tf
9.6862 0 0 9.6862 438.4788 659.7479 Tm
.1678 Tc
(,Y)Tj
/F9 1 Tf
6.7803 0 0 6.7803 448.4079 658.2949 Tm
0 Tc
(P)Tj
/F6 1 Tf
9.6862 0 0 9.6862 454.8669 659.7479 Tm
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(,K)Tj
/F9 1 Tf
6.7803 0 0 6.7803 467.3979 658.2949 Tm
0 Tc
(L)Tj
/F6 1 Tf
9.6862 0 0 9.6862 473.1959 659.7479 Tm
.1678 Tc
(,K)Tj
/F9 1 Tf
6.7803 0 0 6.7803 485.7279 658.2949 Tm
0 Tc
(R)Tj
/F7 1 Tf
9.6862 0 0 9.6862 492.0819 659.7479 Tm
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(\)=)Tj
/F10 1 Tf
-17.3312 -.82 TD
0 Tc
<02>Tj
0 -.8999 TD
<03>Tj
0 -.3 TD
<04>Tj
0 -1.8 TD
<03>Tj
0 -.3 TD
<05>Tj
/F7 1 Tf
.8889 2.3519 TD
(\()Tj
/F6 1 Tf
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(x)Tj
/F5 1 Tf
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<8a>Tj
/F6 1 Tf
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(B)Tj
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6.7803 0 0 6.7803 361.3067 641.1699 Tm
(L)Tj
/F7 1 Tf
9.6862 0 0 9.6862 367.1057 642.6219 Tm
(\))Tj
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<02>Tj
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(K)Tj
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6.7803 0 0 6.7803 388.2478 641.1699 Tm
(L)Tj
/F7 1 Tf
9.6862 0 0 9.6862 396.1978 642.6219 Tm
(+)Tj
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(Y)Tj
/F11 1 Tf
6.7803 0 0 6.7803 413.6598 646.1379 Tm
(0)Tj
/F9 1 Tf
-.3174 -.9119 TD
(P)Tj
/F6 1 Tf
9.6862 0 0 9.6862 446.6989 642.6229 Tm
.2787 Tc
(xTj
/F6 1 Tf
1.0001 0 TD
(B)Tj
/F9 1 Tf
6.7803 0 0 6.7803 361.3067 627.2219 Tm
(R)Tj
/F7 1 Tf
9.6862 0 0 9.6862 367.6617 628.6749 Tm
(\))Tj
/F5 1 Tf
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<02>Tj
/F6 1 Tf
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(K)Tj
/F9 1 Tf
6.7803 0 0 6.7803 388.8038 627.2219 Tm
(R)Tj
/F7 1 Tf
9.6862 0 0 9.6862 397.3098 628.6749 Tm
(+)Tj
/F6 1 Tf
1.0001 0 TD
(Y)Tj
/F9 1 Tf
6.7803 0 0 6.7803 414.7718 632.1899 Tm
(N)Tj
-.3174 -.9119 TD
(P)Tj
/F6 1 Tf
9.6862 0 0 9.6862 446.6989 628.6749 Tm
(x)Tj
/F5 1 Tf
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<03>Tj
/F6 1 Tf
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(B)Tj
/F9 1 Tf
6.7803 0 0 6.7803 472.4959 627.2219 Tm
(R)Tj
/F7 1 Tf
9.6862 0 0 9.6862 332.8186 614.7269 Tm
(\()Tj
/F6 1 Tf
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(x)Tj
/F5 1 Tf
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<8a>Tj
/F6 1 Tf
1.0001 0 TD
(B)Tj
/F9 1 Tf
6.7803 0 0 6.7803 361.3067 613.2739 Tm
.0005 Tc
(idx)Tj
/F7 1 Tf
9.6862 0 0 9.6862 372.9567 614.7269 Tm
0 Tc
(\))Tj
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<02>Tj
/F6 1 Tf
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(K)Tj
/F9 1 Tf
6.7803 0 0 6.7803 394.0988 613.2739 Tm
.0005 Tc
(idx)Tj
/F7 1 Tf
9.6862 0 0 9.6862 407.9008 614.7269 Tm
0 Tc
(+)Tj
/F6 1 Tf
.9999 0 TD
(Y)Tj
/F9 1 Tf
6.7803 0 0 6.7803 425.3628 618.2408 Tm
.0005 Tc
(idx)Tj
-.3175 -.9118 TD
0 Tc
(P)Tj
/F6 1 Tf
9.6862 0 0 9.6862 446.6989 614.7269 Tm
(B)Tj
/F9 1 Tf
6.7803 0 0 6.7803 454.0459 613.2739 Tm
(L)Tj
/F5 1 Tf
9.6862 0 0 9.6862 462.5349 614.7269 Tm
<04>Tj
/F6 1 Tf
1.0555 0 TD
.2787 Tc
(xTj
/F9 1 Tf
.8929 0 TD
(B)Tj
/F12 1 Tf
4.8431 0 0 4.8431 523.714 573.6528 Tm
(L)Tj
ET
.387 w
q 1 0 0 1 500.474 573.071 cm 0 0 m
28.352 0 l
S
Q
BT
/F9 1 Tf
6.7803 0 0 6.7803 511.219 567.3098 Tm
(N)Tj
/F2 1 Tf
9.6862 0 0 9.6862 300.2936 559.0267 Tm
[(denoting)-325.9(the)-326(interv)25(al)-327(length,)-345(these)-326(v)25(alues)-326(are)-326(computed)-327(as)]TJ
0 -1.2001 TD
[(follo)25(ws:)]TJ
/F6 1 Tf
7.5367 -2.3978 TD
.0005 Tc
(idx)Tj
/F7 1 Tf
1.7143 0 TD
0 Tc
(=)Tj
/F5 1 Tf
1.0556 0 TD
<05>Tj
/F6 1 Tf
.5645 .6765 TD
(x)Tj
/F5 1 Tf
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<8a>Tj
/F6 1 Tf
1.0001 0 TD
(B)Tj
/F9 1 Tf
6.7803 0 0 6.7803 430.3148 529.2767 Tm
(L)Tj
ET
q 1 0 0 1 405.594 526.599 cm 0 0 m
30.52 0 l
S
Q
BT
/F6 1 Tf
9.6862 0 0 9.6862 418.3318 517.5327 Tm
(d)Tj
/F5 1 Tf
1.9558 .686 TD
<06>Tj
/F2 1 Tf
8.4057 0 TD
(\(2\))Tj
/F6 1 Tf
-15.5356 -1.926 TD
(B)Tj
/F9 1 Tf
6.7803 0 0 6.7803 375.5617 504.0697 Tm
.0005 Tc
(idx)Tj
/F7 1 Tf
9.6862 0 0 9.6862 389.9008 505.5216 Tm
0 Tc
(=)Tj
/F6 1 Tf
1.0556 0 TD
(B)Tj
/F9 1 Tf
6.7803 0 0 6.7803 407.4728 504.0697 Tm
(L)Tj
/F7 1 Tf
9.6862 0 0 9.6862 415.4238 505.5216 Tm
(+)Tj
/F6 1 Tf
1 0 TD
.0005 Tc
(idx)Tj
/F5 1 Tf
1.6588 0 TD
0 Tc
<02>Tj
/F6 1 Tf
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(d)Tj
/F2 1 Tf
7.2807 0 TD
(\(3\))Tj
/F6 1 Tf
-15.6264 -2.3577 TD
(K)Tj
/F9 1 Tf
6.7803 0 0 6.7803 375.5617 481.2306 Tm
.0005 Tc
(idx)Tj
/F7 1 Tf
9.6862 0 0 9.6862 389.9017 482.6846 Tm
0 Tc
(=)Tj
/F6 1 Tf
1.1756 .6839 TD
(Y)Tj
/F9 1 Tf
6.7803 0 0 6.7803 409.0648 493.4516 Tm
.0005 Tc
(idx)Tj
/F11 1 Tf
1.6465 0 TD
0 Tc
(+1)Tj
/F9 1 Tf
-1.9641 -1.0309 TD
(P)Tj
/F5 1 Tf
9.6862 0 0 9.6862 432.6738 489.3086 Tm
<8a>Tj
/F6 1 Tf
1 0 TD
(Y)Tj
/F9 1 Tf
6.7803 0 0 6.7803 450.1359 492.8226 Tm
.0005 Tc
(idx)Tj
-.3175 -.9118 TD
0 Tc
(P)Tj
ET
q 1 0 0 1 401.289 485.106 cm 0 0 m
60.496 0 l
S
Q
BT
/F6 1 Tf
9.6862 0 0 9.6862 429.0158 476.0396 Tm
(d)Tj
/F2 1 Tf
9.2584 .686 TD
(\(4\))Tj
-21.3477 -2.392 TD
[(W)40(ith)-250(this)-250(de\223nition,)-250(PWLU)-250(has)-250(se)25(v)15(eral)-250(good)-250(properties:)]TJ
/F5 1 Tf
-.2 -1.9393 TD
<80>Tj
/F2 1 Tf
1 0 TD
[(As)-368(a)-368(uni)25(v)15(ersal)-367(approximator)40(,)-398(PWLU)-367(can)-368(closely)-368(ap-)]TJ
0 -1.2 TD
[(proximate)-250(an)15(y)-250(contiguous,)-250(bounded)-250(scalar)-250(functions.)]TJ
/F5 1 Tf
-1 -1.9697 TD
<80>Tj
/F2 1 Tf
1 0 TD
[(PWLU)-898(changes)-899(contiguously)-898.9(with)-899(its)-899.1(parame-)]TJ
0 -1.2 TD
-.0002 Tc
[(ters\(e)14.9(xcept)-351.2(for)-352.2(hyperparameter)]TJ
/F6 1 Tf
12.8618 0 TD
0 Tc
(N)Tj
/F2 1 Tf
.9125 0 TD
[(\),)-377(which)-351(is)-351(friendly)]TJ
-13.7744 -1.2 TD
[(for)-249.9(gradient-based)-249.9(optimization.)]TJ
/F5 1 Tf
-1 -1.9696 TD
<80>Tj
/F2 1 Tf
1 0 TD
[(Most)-245.9(of)-246(the)-247<9465>15(xibilities)-246(lie)-246(in)-246(a)-246(bounded)-247(re)15(gion,)-247(which)]TJ
0 -1.2001 TD
[(can)-250(maximize)-250(the)-250(utilization)-249.9(of)-250(learnable)-250(parameters.)]TJ
/F5 1 Tf
-1 -1.9696 TD
<80>Tj
/F2 1 Tf
1 0 TD
[(Thanks)-291.9(to)-292(the)-292(uniform)-293(di)25(vision)-292(of)-292(interv)25(als,)-303(PWLU)-292(is)]TJ
0 -1.2 TD
[(ef)25(\223cient)-302(in)-302(computation,)-316(especially)-302(for)-302(inference.)-467(W)80(e)]TJ
T*
[(will)-250(address)-249.9(this)-250.1(point)-250(in)-250(Sec)]TJ
/Cs6 cs .92941 .1098 .14118 scn
11.778 0 TD
(4.3)Tj
/Cs8 cs 1 scn
1.2499 0 TD
(.)Tj
/F1 1 Tf
10.6548 0 0 10.6548 300.2936 294.9733 Tm
[(3.2.)-250(Gradients)-250(of)-250(the)-250(Piecewise)-250(Linear)-250(Unit)]TJ
/F2 1 Tf
9.6862 0 0 9.6862 311.9166 277.5372 Tm
[(As)-269(can)-269(be)-268(seen)-269(in)-269(the)-269(pre)25(vious)-269(section,)-273(the)-269(forw)10(ard)-269(pass)]TJ
-1.2 -1.2 TD
[(of)-348(PWLU)-348(is)-349(mostly)-348(dif)25(ferentiable,)-373(which)-348(means)-348(it)-348(can)-349(be)]TJ
T*
[(directly)-274.9(optimized)-275(by)-275(gradients.)-385(W)80(e)-275(deri)25(v)15(e)-275(the)-275(gradients)-275(in)]TJ
T*
[(T)80(able)]TJ
/Cs6 cs .92941 .1098 .14118 scn
2.447 0 TD
(1)Tj
/Cs8 cs 1 scn
.5 0 TD
(.)Tj
-1.747 -1.2 TD
[(Note)-501.9(that)-502(in)-502(the)-503(deri)25(v)25(ation)-502(we)-502(treat)]TJ
/F6 1 Tf
15.9069 0 TD
.0005 Tc
(idx)Tj
/F2 1 Tf
1.9387 0 TD
0 Tc
[(as)-502(constant)]TJ
-19.0456 -1.2001 TD
[(and)-381(omit)-381(related)-380.9(gradients.)-703(F)15(o)0(r)-382(gradients)-382(with)-381(respect)-381(to)]TJ
/F6 1 Tf
T*
(Y)Tj
/F9 1 Tf
6.7803 0 0 6.7803 305.9166 206.3441 Tm
(P)Tj
/F2 1 Tf
9.6862 0 0 9.6862 312.3756 207.7961 Tm
[(,)-399(w)0(e)-369(separate)-370(it)-369(into)-369(four)-370(parts)-369(for)-369(clarity)65(.)-668(When)]TJ
/F6 1 Tf
20.7294 0 TD
(x)Tj
/F5 1 Tf
1.0702 0 TD
<07>Tj
/F7 1 Tf
-23.047 -1.2 TD
([)Tj
/F6 1 Tf
.2778 0 TD
(B)Tj
/F9 1 Tf
6.7803 0 0 6.7803 310.3306 194.72 Tm
.0005 Tc
(idx)Tj
/F6 1 Tf
9.6862 0 0 9.6862 321.9806 196.173 Tm
.1678 Tc
(,B)Tj
/F9 1 Tf
6.7803 0 0 6.7803 333.6326 194.72 Tm
.0005 Tc
(idx)Tj
/F11 1 Tf
1.6465 0 TD
0 Tc
(+1)Tj
/F7 1 Tf
9.6862 0 0 9.6862 355.0897 196.173 Tm
(])Tj
/F2 1 Tf
.5736 0 TD
(where)Tj
/F7 1 Tf
2.7388 0 TD
(0)Tj
/F5 1 Tf
.8627 0 TD
<04>Tj
/F6 1 Tf
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