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WL ComponentFourier components of uTMLP(a,b) or uTMLP(a, b) FVE
cos (W14C)44.6cos(w14a) cos(w14b)- 43.6 sin(w14a) sin(w14b)~ 44.1 cos (w14(a + b))93.2%
sin (w14c)44.1 sin(w14a) cos(w14b) + 44.1 cos(w14a) sin(w14b) ~ 44.1 sin (w14(a + b))93.5%
cos (W35C)40.7 cos(w35a) cos(w35b)- 43.6 sin(w35a) sin(w35b) ~ 42.2 cos (w35(a + b))96.8%
sin (w35C)41.8 sin(w35a) cos(w35b) + 41.8 cos(w35a) sin(w35b) ~ 41.8 sin (w35(a + b))96.5%
cos (w41c)44.8 cos(w41a) cos(w41b) - 44.8 sin(w41a) sin(w41b) ~ 44.8 cos (w41(a + b))97.0%
sin (w41c)44.5 sin(w41a) cos(w41b) + 44.5 cos(w41a) sin(w41b) ~ 44.5 sin (w41(a + b))97.0%
cos (W42C)64.6 cos(w42a) cos(w42b) - 68.5 sin(w42a) sin(w42b) ~ 66.6 cos (w42(a + b))96.4%
sin (w42c)67.8 sin(w42a) cos(w42b) + 67.8 cos(w42a) sin(w42b) ~ 67.8 sin (w42(a + b))96.4%
cos (W52C)60.5 cos(w52a) cos(w52b) - 65.5 sin(w52a) sin(w52b) ~ 63.0cos (w52(a + b))97.4%
sin (w52C)64.5 sin(w52a) cos(w52b) + 64.5 cos(w52a) sin(w52b) ~ 64.5 sin (w52(a + b))98.2%
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WL Component Fourier components of u MLP(a,b) or uT MLP(a,b) FVEcos(w2c) 147.4cos (w2a) cos (w2b)-145.8sin(w2a)sin (w2b) ~ 146.6cos (w2(a +b)) 99.2%sin(w2c) 145.5 cos (wza) sin (w2b) + 145.6 sin(w2a) cos (w2b) ≈ 145.5 sin (w2(a + b)) 99.1%cos (wgc) 49.3cos(wga) cos(wgb) - 48.0 sin (wga) sin (wgb) ~ 48.6 cos (wg(a + b)) 96.4%sin (wgc) 48.6 cos (wga) sin(wgb) + 48.5 sin(wga) cos (wgb) ~ 48.5 sin(wg(a + b)) 96.7%
WL ComponentFourier components of u MLP(a,b) or uT MLP(a,b)oruMLP(a,b)
COSW2C147.4cosw2b)≈146.6cosw2(a+b))
sinW2C145.5 cosw2a) sin(w2b)+145.6sin(w2a) cos(w2b)~145.5sinw2(a+b))
COSwgC49.3cos49.3cos(wga) cos(wgb) - 48.0 sin (wga) sin (wgb) ~ 48.6 cos (wg(a + b))48.6 cos (wga) sin(wgb) + 48.5 sin(wga) cos (wgb) ~ 48.5 sin(wg(a + b))
sinwgC48.6cosgb)≈48.5sin(wg(a+b))
COSW19c)58.0cos (wiga) cos (w1gb)- 58.3 sin (wiga) sin(w1gb) ~ 58.2 cos(w19(a + b))59.3 cos (w1ga) sin (w1gb) + 59.4 sin (w19a) cos (w19b) ~ 59.4sin(w1g(a + b))94.4cos(w31a) cos (w31b) - 96.4sin (w31a) sin(w3ib) ~ 95.4 cos (w31(a + b))97.2 cos (w31a) sin(w31b) + 97.1 sin (w31a) cos (w3ib) ~ 97.2 sin (w31(a +b)58.0 cos(w
sin(w19c)
sin (w31c)cos (w31c)sin (w31c)
WL Component(a) Seed 1
Fourier components of uTMLP(a, b) or uTMLP(a, b)
COSW40C97.0cOsw40aCOs(w40b)-99.4sin(w4oasinW40b) ~ 98.2 cos (w40(a + b))97.3%92.7%
sinW40C81.3cosw40asinw40b+81.3sin(w4oa)COSw40b)≈ 81.3sin(w4o(a+b))
COSW44C309.1cos(w44a)COsW44b-338.7 sin(w44asins (w44a) sin(w44b) + 327.2 sin (w44a) cos (w44b) ~ 327.3 sin (w44(a + b))))
sinW44C327.3cosW44asin (W44b)+ 327.2 sinw44aCos
COSW53C192.1cos (W53a)COSW53b-192.2 sin(w53asin97.3%95.7%FVE
sinW53C166.7cos(w53asin (W53b+ 166.8 sinw53aCos
WL Component(b) Seed 2
WL ComponentFouriercomponents of uMLPa,borUjMLP(a,b)
COsW31C156.1cos(w31a)cosw31b-156.5 sin(w31asin(w31b~156.3ccs(w31(a+b))99.3%98.9%
sinW31C150.7cos(w31asin (w31b)+ 150.7sinw31aCos(w31b~150.7sin(w31(a+b))
COSW45C72.5 cosW45aW45b)-76.8sinW45asinW45b~ 74.6cos(W45(a+b))95.9%
sinW45C74.7cOsw45asinw45b+74.6sin(w45a)COSw45b~ 74.6sinw45(a+b))96.6%
cos(W49C)45.9 cosw49acosw49b)-45.5sinW49a)sinw49b~45.7cos(w49(a+b))97.0%
sinW49C45.8 cosw49asinw49b+45.8sin(w49a)COsW4gb~ 45.8sinw49(a+b))96.9%
cos((W52C)sin (w52C)71.6cosw52acos(w52b)-72.1sin(W52qsin (W52b~ 71.9 cosw52(a+b))w52(a +b)
68.7cosW52asinW52b)+68.7sin(W52a)CosW52b~ 68.7sin
(c) Seed 3FVE
WLCWL ComponentFouriercomponentsof uMLP(a,b)orvMLP(a,b)
cossin(w17C)cos (w32c)sin(w32c)cos (W42C)(W17C)66.0cosw17acosw17b)-63.5sin(sin(w17b~64.8cosw17(a+b))96.4%94.9%96.2%
W17C66.4cosw17asinw17b)+66.4sin(w17a)Cosw17b~ 66.4sinw1z(a+b))
W32C68.7cosW32acos(w32b-68.4sin(Wa2q)sin (.68.5.cosWaa(a +b))
W32C68.0 cosw32asinW32b+68.0sin(W32a)CosW32b~ 68.0sinw32(a+b))96.3%
W42C100.4cos(w420)cos(w42)-96.0sin(w42asinw42b~98.2cos(w42(a+b))97.9%
W42C100.2 cos (W42a)sin (w42b)+ 100.1 sin(w42gcos(w42b~100.1sii(w42(a+b))98.6%
COSW51C118.0 cos (w51a)CosW51b-116.2 sin(w51asinW51b≈117.1cos(w51(a+b))99.0%
sinW51C114.3cos (w51a)sin (w51b)+ 114.2 sinw51acos(w51b)≈114.2sin(w51(a+b))98.5%
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SeedTest LossLoss (Key frequencies removed)|Loss (All other frequencies removed)
12.07:10-76.5:1005.7.10-8
22.1·10-71.1 · 1016.2.10-8
32.05·10-76.7·1005.5.10-8
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ModelTest LossGini(Wε)Gini(WL)Key FrequenciesLogit FVEMLP FVE
40% Training Data1.98:10-70.760.79[17,43,49,55]94.9%83.3% [26.1%]
50% Training Data1.68 : 10-70.750.77[2,17,31,41, 44]91.2%85.2% [28.2%]
60% Training Data1.23.10-70.790.84[2,23,34,51]96.4%95.7% [1.4%]
70% Training Data9.85.10-80.800.91[14,15,26]99.0%98.9% [0.4%]
80% Training Data5.83.10-70.620.80[38,41]63.9%94.1% [2.5%]
90% Training Data1.11:10-70.790.88[3,26,34,43]98.6%98.7% [0.3%]
2Layer Transformer9.54:10-70.590.80[14,18,29]91.8%95.2%[1.9%]
2 Layer Transformer4.41:10-50.550.73[7,12,35,49]86.1%86.2% [6.4%]
2 Layer Transformer6.50.10-20.660.80[4,9,28]88.5%85.4% [5.9%]
2 Layer Transformer4.18.10-20.560.76[4,5,15,54]91.4%81.2% [17.8%]
2 Layer Transformer1.75:10-20.680.71[3,4,13,30,38]84.0%71.9% [19.5%]
P=533.00:10-40.610.68[6,9,16,21]91.2%90.2% [5.8%]
P=531.03:10-40.560.72[4,13,16]94.8%93.1% [6.4%]
P=531.21:10-50.660.79[13,22,23]98.2%97.6% [0.9%]
P=533.95·10-60.660.74[3,14,15]88.5%91.8% [4.6%]
P=535.56:10-60.670.80[10,14,22]98.1%98.3% [0.6%]
P=1092.02:10-70.760.83[6,7,22,25]98.0%97.3% [1.9%]
P=1092.95.10-70.690.82[8,14,29,32,41]95.2%94.7% [2.3%]
P=1091.66·10-70.780.86[13,23,39,45]98.5%97.6% [0.9%]
P=1092.50.10-70.680.82[8,13,32,41]96.8%95.5% [2.3%]
P=1092.77:10-70.760.85[29,37,38,49]97.9%98.1% [0.8%]
Dropout p= 0.22.65:10-10.190.46[1,4,7,17,22,33,40,49,55]71.3%65.0% [17.5%]
Dropout p = 0.24.52.10-10.190.46[3,8,19,28,32,34,40,44]73.3%71.4% [10.7%]
Dropout p = 0.22.03:10-10.200.45[4,5,32,38,41,44,49,50]74.2%71.1% [10.6%]
Dropout p = 0.5<10-80.260.56[1,4,26,46,47,55]89.4%88.9%[3.5%]
Dropout p = 0.52.01.10-20.200.49[16,21,35,47,53]88.4%88.4% [3.0%]
Dropout p = 0.5<10-80.250.54[1,4,7,19,29,31,42]86.1%85.6%[4.0%]
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Model TypeLossAccuracyAblated LossAblated Acuracy
VaryingData Fraction1.83:10-7 (1.65:10 7100%7.74· 10-7 (6.74· 10-7)100%
2 Layer Transformer1.97·10-2 (2.41·10-299.6%4.63· 10-2 (6.72 : 10-298.7%
P=535.96·10-5 (8.91·10-5)100%1.5·10-4 (2.70· 10-4)100%
P=1091.94· 10-7 (3.74· 10-8)100%6.53· 10-7 (1.41 · 10-7)100%
Dropout p = 0.20.215 (0.091)99.7%0.205 (0.075)99.7%
Dropout p = 0.54.68 · 10-³ (8.11· 10-3)100%3.6:10-3 (5.82.10-3)100%
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