# Controlling the overlap of first-layer weights in 2LNN in the meanfield regime

## Nov 2020

```python
%matplotlib inline
%config InlineBackend.figure_format = 'retina'

import numpy as np
import numpy.random as rnd

import matplotlib as mpl
import matplotlib.pyplot as plt

plt.rc("font", **{"size": 14})
plt.rc("lines", linewidth=2.0)
mpl.rcParams['lines.markersize'] = 12
mpl.rcParams['lines.markeredgewidth'] = 2
```

We propose a measure for the overlap of the first layer weights of two-layer neural nets $\phi_\theta: \mathbb{R}^N \to \mathbb{R}$ in the meanfield regime:
\begin{equation}
  \phi_\theta(x) = \frac{1}{K} \sum_k^K v^k g(w^k x)
\end{equation}
where $\theta$ denotes all the trainable parameters of the model.

We first draw a weight matrix for the first teacher $W_1=(w_1^k)\in\mathbb{R}^{N\times K}$ element-wise i.i.d. from the normal distribution and orthonormalise it. The first-layer weights for the second teacher $W_2$ are obtained from
\begin{equation}
  W_2 = \alpha W_1 + (1 - \alpha) Z
\end{equation}
where $Z$ is another matrix with i.i.d. Gaussian elements and $0<\alpha<1$ is an interpolation parameter. We now compute the overlap of the two weight matrices for different values of $\alpha$.

```python
N = 15  # input dimension
K = 1000  # num of neurons in the hidden layer
```

```python
alphas = np.linspace(0, 1)  # interpolation parameter
```

We will calculate the mean and standard deviation of two overlap matrices:
\begin{equation}
  \frac{W W^\top}{K} \qquad \mathrm{and}\qquad \frac{W^\top W}{N}
\end{equation}

```python
means = {"wwT": [None] * len(alphas), "wTw": [None] * len(alphas)}
stds = {"wwT": [None] * len(alphas), "wTw": [None] * len(alphas)}
```

```python
for idx, alpha in enumerate(alphas):
    ws = [rnd.randn(N, K), None]  # first layer of weights

    # orthonormalise the weights of the first teacher;
    # this reduces the fluctuations in the overlap measured below
    import pdb; pdb.set_trace()
    overlap = ws[0] @ ws[0].T / K
    L = np.linalg.cholesky(overlap)
    ws[0] = np.linalg.inv(L) @ ws[0]

    import pdb; pdb.set_trace()

    # Compute the first-layer weights of the second teacher
    ws[1] = alpha * ws[0]  + (1 - alpha) * rnd.randn(*ws[0].shape)

    # calculate diagonal elements of overlaps
    overlaps = {"wwT": np.diag(ws[0] @ ws[1].T / K),
                "wTw": np.diag(ws[0].T @ ws[1] / N)}
    for key in ["wwT", "wTw"]:
        means[key][idx] = np.mean(overlaps[key])
        stds[key][idx] =  np.std(overlaps[key])
```

    > <ipython-input-14-392986fe2a65>(7)<module>()
          5     # this reduces the fluctuations in the overlap measured below
          6     import pdb; pdb.set_trace()
    ----> 7     overlap = ws[0] @ ws[0].T / K
          8     L = np.linalg.cholesky(overlap)
          9     ws[0] = np.linalg.inv(L) @ ws[0]

    ipdb>  overlap.shape

    (15, 15)

    ipdb>  n

    > <ipython-input-14-392986fe2a65>(8)<module>()
          6     import pdb; pdb.set_trace()
          7     overlap = ws[0] @ ws[0].T / K
    ----> 8     L = np.linalg.cholesky(overlap)
          9     ws[0] = np.linalg.inv(L) @ ws[0]
         10

    ipdb>  n

    > <ipython-input-14-392986fe2a65>(9)<module>()
          7     overlap = ws[0] @ ws[0].T / K
          8     L = np.linalg.cholesky(overlap)
    ----> 9     ws[0] = np.linalg.inv(L) @ ws[0]
         10
         11     # Compute the first-layer weights of the second teacher

    ipdb>  n

    > <ipython-input-14-392986fe2a65>(12)<module>()
         10
         11     # Compute the first-layer weights of the second teacher
    ---> 12     ws[1] = alpha * ws[0]  + (1 - alpha) * rnd.randn(*ws[0].shape)
         13
         14     # calculate diagonal elements of overlaps

    ipdb>  we[0]

    *** NameError: name 'we' is not defined

    ipdb>  ws[0]

    array([[ 0.98429131, -2.67154787, -0.24771118, ..., -1.68002211,
             0.81149517,  0.96944716],
           [ 1.41339722, -0.29548129,  0.92572614, ..., -0.20880481,
            -0.87499458, -1.7516088 ],
           [-1.2827928 , -1.35265813, -1.07746497, ...,  1.13630705,
            -1.88459379, -0.37835771],
           ...,
           [-0.41824515,  1.63803122,  0.86109333, ..., -0.81629009,
             0.50190841,  0.86265532],
           [ 0.81372469,  0.54548931, -0.28754634, ...,  0.61628861,
            -1.47490567, -0.59947689],
           [ 0.0220776 , -0.33616221,  0.7050174 , ...,  0.19256498,
            -0.20096568,  0.06318821]])

    ipdb>  ws[0].shape

    (15, 1000)

    ipdb>  ws[0][0]

    array([ 9.84291314e-01, -2.67154787e+00, -2.47711178e-01, -1.02029636e+00,
           -1.65410677e-01, -3.89257516e-01, -1.43501747e-01, -1.36789983e-02,
            1.11712103e+00,  3.73601046e-01, -2.39889307e-01, -9.41772074e-01,
           -2.72733437e+00,  1.90295545e+00,  7.78069913e-01,  6.15355372e-01,
            5.02227079e-01,  5.95148220e-01, -6.74557003e-01, -1.67540061e+00,
           -3.88751246e-01,  7.25295361e-01, -1.56139492e+00, -1.74060487e+00,
            4.27186242e-01, -3.91127127e-01, -2.70114437e-01, -1.03652805e-01,
            1.28952992e-01, -3.57169044e-01, -1.07204561e+00,  2.08143021e+00,
            5.44215711e-01, -3.68947001e-01,  9.92032509e-01,  4.65406769e-01,
            1.11568972e+00, -8.01249721e-02, -1.39841164e+00,  1.15093542e-01,
            1.50016540e+00,  1.00177466e+00,  1.05624218e-02,  5.71041762e-01,
           -1.19683936e+00,  1.19632552e+00,  9.66264547e-01,  1.25781920e+00,
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           -3.26935662e-01, -3.17915774e-01,  1.23800377e+00,  9.23184946e-01,
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            2.28218814e+00,  8.51137604e-01,  7.31947502e-01, -2.32982144e+00,
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           -2.51232959e+00, -1.21331210e-03, -1.21410687e-01,  6.67710798e-01,
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           -2.30855610e+00,  3.31771788e-01,  1.18598200e+00,  4.02037759e-01,
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            3.80007901e-01, -6.38946480e-01,  1.20334841e+00, -8.34228630e-01,
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           -5.82191967e-03,  1.11204014e+00, -1.39568191e-01, -9.01259741e-02,
           -2.79762097e-01,  1.03063313e+00,  8.48560955e-01,  3.64627721e-03,
           -1.00023032e+00, -6.53885466e-01,  2.08334537e+00,  1.12769697e+00,
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           -1.55615354e+00, -1.40937502e+00, -7.37929538e-01, -2.62707286e-01,
            7.61765279e-01,  2.94658958e-01,  2.67495084e-01, -6.20078767e-01,
           -1.22010774e+00,  2.29993209e-01,  8.49816196e-02,  5.90966841e-01,
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           -1.24312598e+00,  2.48972585e-01, -1.14381544e-01, -8.27526395e-01,
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           -7.69877163e-01, -3.25454334e-01, -1.65429240e+00, -1.37599026e+00,
           -1.02267312e-01,  7.42555341e-01,  1.82670387e-01, -7.44059339e-01,
            1.92142048e+00, -4.48962982e-01,  5.73471690e-01,  1.59425925e+00,
            7.94458573e-01,  1.60956234e-01, -1.82155993e+00, -2.94798391e-01,
            1.42499421e+00,  6.40541912e-01, -8.58036622e-01,  5.01906348e-02,
            2.10341824e-01,  2.27829896e-01,  1.73965430e-01,  2.13025461e+00,
           -7.26232261e-01, -3.37701641e-01,  4.30992835e-01, -5.89017127e-01,
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           -5.77852962e-01, -6.98284365e-01,  1.61031679e-01,  4.45699637e-01,
           -1.21426278e+00,  1.16179650e+00, -1.16362782e+00,  2.95560041e-01,
           -1.77993501e+00, -6.23721615e-01,  1.19663374e+00, -5.12522002e-01,
           -6.79025785e-01,  4.73970927e-03,  2.36189587e-02, -1.09738120e+00,
           -1.81525789e+00,  1.04241712e+00, -2.93739211e-01,  8.83260632e-01,
           -1.60817534e-01, -7.66919955e-01, -4.38373244e-02, -3.50647359e-02,
            4.79043236e-01,  1.79482102e+00, -1.09083112e-01, -2.75494498e-01,
           -1.11406559e+00, -2.75079324e-01, -6.23134082e-01,  3.04734162e-01,
            1.28850973e+00, -2.83485108e-01, -1.69655630e+00,  9.79401597e-01,
           -3.12492901e-01,  2.94454403e-01, -1.26683958e+00,  4.46599200e-01,
           -2.93666269e-01,  1.58311654e+00,  9.87463975e-01, -1.10681478e+00,
            7.95284927e-01, -5.82788181e-01,  2.27962712e-01,  1.04370482e+00,
            8.23308661e-01, -1.51070222e+00, -2.18168940e-01, -9.81920417e-01,
           -1.45499130e+00, -3.92751781e-01,  1.25545699e-01, -9.90084376e-01,
            1.00116657e+00,  2.47838167e-01,  3.00519072e-01, -2.82124234e-01,
            4.17485530e-01,  3.18968275e-01,  2.96229723e-01, -4.17821942e-01,
           -1.45871631e+00,  1.62479699e-01, -8.56423605e-03,  3.96696834e-01,
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           -1.98153662e+00, -8.74845568e-01,  1.87622408e-01, -6.15385817e-02,
            2.76841035e-01,  1.66629000e-01,  1.79155535e-01, -9.03926259e-01,
           -7.24157936e-01,  7.93521710e-01, -1.64952810e+00, -1.70205072e+00,
            2.05825595e-01,  7.40382379e-01, -5.37714462e-01, -1.52694761e+00,
            6.32964258e-01,  4.14327408e-01, -3.11672566e-01,  2.78290256e-01,
            5.39156542e-01,  2.83981209e-01,  9.59175834e-02, -1.42824551e+00,
           -5.39643678e-01, -8.76655697e-01, -1.03197996e+00, -8.68542043e-01,
           -1.12500399e+00,  1.90393480e+00,  2.65147680e+00, -5.59393893e-01,
            8.56445023e-02,  7.26434508e-01, -2.72738158e-01,  9.95835144e-01,
           -1.67793063e-01, -1.77687724e-01, -8.06109704e-01, -1.24820053e+00,
           -6.33703578e-01,  2.24870569e+00,  1.29512088e+00, -9.11925437e-01,
           -5.74153672e-01,  1.63023433e+00, -7.19184233e-01,  1.90290227e-01,
            4.22116506e-01, -9.23919995e-01,  1.11689852e+00,  1.33079877e+00,
           -4.92987846e-01,  6.88298610e-01, -1.03686606e+00, -1.12047763e+00,
           -8.23447240e-01,  3.43245350e-01, -8.78207308e-01,  1.61951865e+00,
           -1.97437355e+00,  1.75049893e-01,  1.88049874e-01,  8.59878182e-01,
           -1.05445976e+00, -1.22640363e+00, -7.78134507e-01, -3.76029154e-01,
            1.99887836e-01,  2.77795070e-02, -1.65239496e+00, -1.11167128e+00,
            1.97934406e+00, -7.14254156e-01,  1.55912743e+00,  1.57221889e+00,
           -9.83464734e-02,  5.78516832e-01,  1.82554545e+00,  2.48215884e-01,
           -7.68486404e-01, -7.53570547e-01,  1.59776693e+00, -1.62883237e-01,
           -1.06246786e+00, -3.67058822e-01,  1.04559196e+00, -2.89745964e-01,
           -2.51721276e+00,  1.22605302e+00,  1.48575112e-01, -2.09257504e+00,
            8.12288670e-01,  1.18754540e+00,  1.02880847e+00,  2.09006018e+00,
           -2.65374169e-01,  2.60460392e-01, -1.29239184e+00,  2.97213013e-01,
            5.54126915e-01, -1.07758083e+00, -4.24177778e-01, -1.09759532e+00,
            5.67577251e-01, -1.20873008e-01,  8.23912142e-01, -8.05800702e-01,
           -1.92109475e+00, -1.22895780e+00,  7.68528239e-01,  1.71249945e+00,
            8.24228574e-02, -7.61190964e-01, -2.72912507e-01,  1.83584542e+00,
            6.72885061e-01, -4.45754189e-01,  1.21704102e+00,  1.92145180e-01,
            1.51088234e-01,  1.59421281e+00, -1.33765095e+00, -4.85955977e-02,
            1.00634056e+00,  4.69892652e-01,  9.07220541e-01,  4.32247353e-01,
           -1.66620371e-01, -8.53670669e-01, -1.22617747e+00,  5.99291659e-01,
           -2.66835590e+00, -4.58305790e-01, -1.12783883e+00, -3.18829542e-01,
            2.78953799e-01, -1.13456906e+00, -2.07971462e-01,  1.22118521e+00,
            4.80748014e-01, -5.01151925e-01, -1.31616443e+00, -6.79367555e-01,
            3.55068652e-01, -2.54682120e-01,  8.84540604e-01, -4.82088103e-01,
           -3.79796475e-01, -6.94715172e-01,  5.42787585e-01, -6.15518676e-01,
            1.33571333e+00, -7.53093053e-01,  1.59464820e-01, -3.42610338e-01,
            5.80974799e-01,  3.28274980e-01, -9.88035155e-01, -1.46675025e+00,
           -9.37054031e-01,  3.38921708e-01, -2.44935891e-01, -7.89146451e-01,
           -3.06348465e-01,  1.46735156e-01,  6.25030298e-01, -5.66648854e-01,
           -2.57801535e-01, -1.01865561e+00, -3.74768680e-01,  2.75270929e+00,
            4.58425841e-01, -3.96170878e-01, -6.38005380e-02, -1.21525840e+00,
           -1.23838271e+00,  2.90287988e-01,  1.14194480e+00,  5.68948251e-02,
           -1.85129600e+00, -2.51033526e-01,  8.25679104e-01, -3.46746831e-01,
            1.90964561e+00,  4.93143689e-01,  5.48691564e-01, -7.70165399e-02,
           -4.21678149e-03, -1.85590098e-01, -5.48877535e-03, -3.10558380e-01,
            2.41220649e+00,  1.50034552e+00,  9.79070079e-01, -5.34708082e-02,
           -2.73486319e-01,  9.03213412e-01,  2.15679497e-01, -1.29718753e+00,
           -9.65043353e-01, -7.79312710e-02, -1.27718813e+00, -1.24837109e+00,
           -3.37617288e-01, -8.73219558e-01, -7.07524465e-01,  4.15596111e-01,
            4.89763712e-01,  4.03658602e-01,  1.09086422e+00, -1.10907954e+00,
            9.10939858e-01,  1.31737298e+00, -5.76893677e-01,  1.20213536e-01,
            1.44081529e+00,  4.15123562e-01, -2.39042389e-01, -5.19599319e-01,
            4.94312960e-01, -2.35442242e+00,  8.63153490e-01, -1.32585306e-02,
            1.33631061e+00, -1.98589487e+00, -9.39927021e-01, -6.45488712e-01,
           -4.46334196e-01, -3.96770210e-01, -1.09963469e+00,  7.20233038e-01,
           -4.09746689e-01,  4.19550541e-01, -9.27710948e-01,  1.86459189e-02,
            4.83423364e-01,  5.42658361e-01,  6.08198560e-02,  2.26481384e+00,
            1.03788850e+00, -1.18080652e+00,  3.56716052e-01, -2.72060736e-01,
           -4.80617606e-01,  1.37980882e+00,  2.28088697e+00,  1.55715620e+00,
            6.30819552e-01,  1.26978350e+00, -5.60972123e-02, -1.65819606e+00,
           -8.47557228e-02, -7.58907663e-01, -1.93866968e+00,  8.07162652e-01,
           -1.72856201e+00, -7.77741973e-01, -1.90586911e-01, -2.79493074e-01,
           -2.41584752e-01, -4.80816958e-01,  5.04670916e-01,  6.47506095e-02,
           -3.17709797e-01,  5.45082790e-01, -9.11238672e-01,  1.03883342e+00,
           -4.06591132e-02,  1.79114512e+00, -5.99343444e-01,  3.67208439e-01,
           -1.36955178e+00, -1.68002211e+00,  8.11495174e-01,  9.69447164e-01])

    ipdb>  sum(ws[0][0])

    -21.479860284119574

    ipdb>  sum(ws[:, 0])

    *** TypeError: list indices must be integers or slices, not tuple

    ipdb>  ws[0][0]

    array([ 9.84291314e-01, -2.67154787e+00, -2.47711178e-01, -1.02029636e+00,
           -1.65410677e-01, -3.89257516e-01, -1.43501747e-01, -1.36789983e-02,
            1.11712103e+00,  3.73601046e-01, -2.39889307e-01, -9.41772074e-01,
           -2.72733437e+00,  1.90295545e+00,  7.78069913e-01,  6.15355372e-01,
            5.02227079e-01,  5.95148220e-01, -6.74557003e-01, -1.67540061e+00,
           -3.88751246e-01,  7.25295361e-01, -1.56139492e+00, -1.74060487e+00,
            4.27186242e-01, -3.91127127e-01, -2.70114437e-01, -1.03652805e-01,
            1.28952992e-01, -3.57169044e-01, -1.07204561e+00,  2.08143021e+00,
            5.44215711e-01, -3.68947001e-01,  9.92032509e-01,  4.65406769e-01,
            1.11568972e+00, -8.01249721e-02, -1.39841164e+00,  1.15093542e-01,
            1.50016540e+00,  1.00177466e+00,  1.05624218e-02,  5.71041762e-01,
           -1.19683936e+00,  1.19632552e+00,  9.66264547e-01,  1.25781920e+00,
            5.39271060e-01,  2.59142814e-01,  2.25129475e-01, -4.19216543e-01,
           -3.26935662e-01, -3.17915774e-01,  1.23800377e+00,  9.23184946e-01,
           -8.37310826e-01, -5.03856664e-02, -9.87949256e-02, -2.99166803e-01,
            2.28218814e+00,  8.51137604e-01,  7.31947502e-01, -2.32982144e+00,
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           -2.17156123e+00, -6.10592540e-01, -3.94995003e-02,  1.43910523e+00,
           -3.31352625e-01,  2.40060782e+00,  1.13188660e-01,  1.38870499e-01,
           -4.55218021e-01, -1.01446149e+00, -4.35811268e-01, -1.64153446e-01,
            5.24502470e-01,  3.39243052e+00,  3.59381107e-01,  3.86981560e-01,
           -2.51232959e+00, -1.21331210e-03, -1.21410687e-01,  6.67710798e-01,
            1.14347721e+00, -5.36425690e-01,  2.17181972e-01, -5.59598350e-01,
            9.75168230e-01, -5.14634205e-01, -5.81507896e-01, -4.77028875e-01,
           -1.74804083e+00,  1.60496408e+00,  1.13275195e+00, -6.95774124e-02,
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           -4.81873905e-01,  1.67054591e-01,  9.35608651e-01, -4.60667543e-01,
           -5.66701133e-01,  2.99783302e-01, -1.40663804e-01,  2.82561049e-01,
           -4.45756576e-01, -1.73019799e+00, -3.21366614e-01,  1.36956498e+00,
            1.66702501e+00,  1.83065173e+00, -1.22085263e-01, -1.36478960e+00,
            1.74504022e-01,  1.75264518e-01, -6.16512548e-01,  1.12353659e+00,
            9.32440990e-01, -2.16105604e-01, -3.33619008e-01, -5.72438865e-01,
            7.50679815e-01, -8.91832022e-01, -5.97205414e-01, -4.74942807e-01,
            9.28396568e-01,  9.56712412e-01, -1.27674585e+00,  7.83552164e-01,
           -8.07671750e-01,  7.06130720e-01,  3.67023768e-01, -5.43778749e-01,
           -3.31812462e-01,  8.65133324e-01, -1.77622194e-01,  5.91300284e-01,
            1.70962732e+00, -1.47078518e+00,  1.20761351e+00,  1.20885814e+00,
           -1.24029244e+00, -6.63599727e-01, -1.46291702e+00, -6.35502592e-01,
            6.84424472e-01, -1.40903283e+00,  1.42180200e-01, -5.71668361e-01,
           -1.85136293e+00, -4.30382095e-01,  1.47271723e-01,  2.23333212e-01,
           -6.89780287e-01,  2.19725763e+00,  2.63307518e+00,  1.97236353e+00,
            8.23191353e-01,  5.56538342e-02, -3.52996919e+00,  1.41044467e+00,
            3.41319542e-01,  5.06460693e-01,  5.79310449e-01,  9.00474515e-01,
           -3.82200019e-01, -8.43958768e-01, -4.69473706e-01, -1.39621511e+00,
            4.36596569e-01,  7.66745499e-01,  4.43824348e-01, -2.65018176e-01,
            8.02742819e-01, -6.35882423e-01, -8.35163118e-01,  1.47156541e+00,
            9.17680154e-02,  1.02333081e+00,  6.79583470e-01,  2.06895951e+00,
            1.69420979e+00, -1.21819770e+00, -6.17625690e-01,  1.57318377e+00,
           -2.20463358e-01, -1.20291032e-01,  1.33435380e+00,  5.16936563e-01,
            6.62461555e-02, -1.77709014e-01, -1.73168418e+00,  6.97699457e-01,
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           -4.00578711e-01,  1.09641699e+00, -3.40321445e-01, -7.23078759e-01,
           -8.36529651e-02,  3.47386140e-01,  1.57557122e+00,  3.56339172e-01,
           -4.41517017e-02, -9.93100739e-01, -3.12641389e-01,  1.40525820e+00,
           -1.25369931e+00, -4.72390451e-01, -8.24372075e-01,  5.74267520e-01,
           -2.70343533e-01, -2.80307854e+00,  1.63429549e+00, -1.10882680e+00,
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           -2.38046775e-01,  1.21565234e+00, -4.34949773e-02, -6.79021269e-01,
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           -5.87084605e-01,  4.30710210e-01,  1.83406642e-01,  1.58691181e-01,
           -8.00968128e-01, -8.24876156e-02,  2.69063414e-01, -9.41818883e-01,
           -8.40046442e-01, -1.15590334e+00,  1.91490394e+00,  1.43777806e+00,
            3.98887051e-01, -1.61165614e+00, -9.58909722e-01, -9.44538606e-01,
           -2.35627765e-01,  4.05319430e-01, -6.24287614e-01,  1.76277038e+00,
           -1.50840950e+00,  6.40593052e-01, -1.98516720e-01,  5.87667293e-01,
            4.94615260e-01, -3.47288224e-01, -8.82136401e-01,  6.41703531e-01,
           -4.30772370e-01,  1.77490642e-01,  9.91991265e-02, -4.31143216e-01,
           -2.84857282e-02,  1.70862789e+00,  4.61032075e-01, -1.41946064e+00,
            5.61488815e-01,  2.65696478e-01, -6.41186330e-01, -6.22852619e-01,
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           -1.36955178e+00, -1.68002211e+00,  8.11495174e-01,  9.69447164e-01])

    ipdb>  ws[0][:, 0]

    array([ 0.98429131,  1.41339722, -1.2827928 , -0.01663013,  0.09914473,
            0.81230053, -0.95214137, -0.27462135,  0.10819478,  0.04855695,
           -0.28010791, -1.28745879, -0.41824515,  0.81372469,  0.0220776 ])

    ipdb>  sum(ws[0][:, 0] ** 2)

    8.851645718160007

    ipdb>  ortho = ws[0]
    ipdb>  ortho[0]

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           -8.49183955e-01, -1.08721884e-01, -4.34838409e-01,  7.77453063e-01,
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           -2.07336957e-01,  2.29240563e+00,  2.06137097e+00,  5.05011673e-01,
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           -6.17413277e-01, -7.53552255e-01,  1.34034007e-01, -1.49411361e-01,
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           -1.62964501e-01,  1.66564685e+00,  2.08629938e+00, -3.69680056e-01,
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            1.92415788e-01, -4.47570285e-01,  1.00519390e+00, -3.15254632e-01,
           -1.43570364e+00,  1.97543301e+00, -2.82892094e-01,  9.07911015e-01,
           -1.00158074e+00, -8.55191374e-02, -2.63000819e+00,  8.42978050e-01,
           -3.29754048e-01,  9.31147872e-01,  1.47523059e-01, -1.04838319e+00,
            3.98612812e-01,  1.45708799e+00,  9.86318777e-01, -1.69579837e+00,
            3.58197765e-01, -6.93540421e-01, -3.21295240e-01, -8.16035280e-01,
           -5.78962023e-01, -4.58552126e-01, -8.19010423e-01, -2.21165699e+00,
            1.89472305e-01,  2.93773550e-01, -2.85262870e-01,  9.02379379e-02,
            5.51425212e-01,  9.17305438e-01, -1.13496529e+00, -2.28384863e+00,
           -1.15920188e+00, -4.06816355e-01,  3.47059470e-01,  1.71493042e-01,
            1.90203558e+00, -1.58472567e+00,  1.74444300e+00,  6.55836929e-01,
           -2.30855610e+00,  3.31771788e-01,  1.18598200e+00,  4.02037759e-01,
           -1.73304195e+00, -1.04305233e+00,  1.22703731e+00, -2.67177603e-01,
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           -5.13853465e-01,  4.34343334e-01,  1.83341759e+00,  6.42530227e-01,
           -5.82191967e-03,  1.11204014e+00, -1.39568191e-01, -9.01259741e-02,
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           -1.24312598e+00,  2.48972585e-01, -1.14381544e-01, -8.27526395e-01,
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           -1.02267312e-01,  7.42555341e-01,  1.82670387e-01, -7.44059339e-01,
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            7.94458573e-01,  1.60956234e-01, -1.82155993e+00, -2.94798391e-01,
            1.42499421e+00,  6.40541912e-01, -8.58036622e-01,  5.01906348e-02,
            2.10341824e-01,  2.27829896e-01,  1.73965430e-01,  2.13025461e+00,
           -7.26232261e-01, -3.37701641e-01,  4.30992835e-01, -5.89017127e-01,
            9.95471147e-01, -3.28638941e-01, -1.91851506e-01, -1.00146658e+00,
           -5.77852962e-01, -6.98284365e-01,  1.61031679e-01,  4.45699637e-01,
           -1.21426278e+00,  1.16179650e+00, -1.16362782e+00,  2.95560041e-01,
           -1.77993501e+00, -6.23721615e-01,  1.19663374e+00, -5.12522002e-01,
           -6.79025785e-01,  4.73970927e-03,  2.36189587e-02, -1.09738120e+00,
           -1.81525789e+00,  1.04241712e+00, -2.93739211e-01,  8.83260632e-01,
           -1.60817534e-01, -7.66919955e-01, -4.38373244e-02, -3.50647359e-02,
            4.79043236e-01,  1.79482102e+00, -1.09083112e-01, -2.75494498e-01,
           -1.11406559e+00, -2.75079324e-01, -6.23134082e-01,  3.04734162e-01,
            1.28850973e+00, -2.83485108e-01, -1.69655630e+00,  9.79401597e-01,
           -3.12492901e-01,  2.94454403e-01, -1.26683958e+00,  4.46599200e-01,
           -2.93666269e-01,  1.58311654e+00,  9.87463975e-01, -1.10681478e+00,
            7.95284927e-01, -5.82788181e-01,  2.27962712e-01,  1.04370482e+00,
            8.23308661e-01, -1.51070222e+00, -2.18168940e-01, -9.81920417e-01,
           -1.45499130e+00, -3.92751781e-01,  1.25545699e-01, -9.90084376e-01,
            1.00116657e+00,  2.47838167e-01,  3.00519072e-01, -2.82124234e-01,
            4.17485530e-01,  3.18968275e-01,  2.96229723e-01, -4.17821942e-01,
           -1.45871631e+00,  1.62479699e-01, -8.56423605e-03,  3.96696834e-01,
            4.18949089e-01, -1.43310801e-01,  7.07014221e-01,  7.01616614e-01,
           -1.98153662e+00, -8.74845568e-01,  1.87622408e-01, -6.15385817e-02,
            2.76841035e-01,  1.66629000e-01,  1.79155535e-01, -9.03926259e-01,
           -7.24157936e-01,  7.93521710e-01, -1.64952810e+00, -1.70205072e+00,
            2.05825595e-01,  7.40382379e-01, -5.37714462e-01, -1.52694761e+00,
            6.32964258e-01,  4.14327408e-01, -3.11672566e-01,  2.78290256e-01,
            5.39156542e-01,  2.83981209e-01,  9.59175834e-02, -1.42824551e+00,
           -5.39643678e-01, -8.76655697e-01, -1.03197996e+00, -8.68542043e-01,
           -1.12500399e+00,  1.90393480e+00,  2.65147680e+00, -5.59393893e-01,
            8.56445023e-02,  7.26434508e-01, -2.72738158e-01,  9.95835144e-01,
           -1.67793063e-01, -1.77687724e-01, -8.06109704e-01, -1.24820053e+00,
           -6.33703578e-01,  2.24870569e+00,  1.29512088e+00, -9.11925437e-01,
           -5.74153672e-01,  1.63023433e+00, -7.19184233e-01,  1.90290227e-01,
            4.22116506e-01, -9.23919995e-01,  1.11689852e+00,  1.33079877e+00,
           -4.92987846e-01,  6.88298610e-01, -1.03686606e+00, -1.12047763e+00,
           -8.23447240e-01,  3.43245350e-01, -8.78207308e-01,  1.61951865e+00,
           -1.97437355e+00,  1.75049893e-01,  1.88049874e-01,  8.59878182e-01,
           -1.05445976e+00, -1.22640363e+00, -7.78134507e-01, -3.76029154e-01,
            1.99887836e-01,  2.77795070e-02, -1.65239496e+00, -1.11167128e+00,
            1.97934406e+00, -7.14254156e-01,  1.55912743e+00,  1.57221889e+00,
           -9.83464734e-02,  5.78516832e-01,  1.82554545e+00,  2.48215884e-01,
           -7.68486404e-01, -7.53570547e-01,  1.59776693e+00, -1.62883237e-01,
           -1.06246786e+00, -3.67058822e-01,  1.04559196e+00, -2.89745964e-01,
           -2.51721276e+00,  1.22605302e+00,  1.48575112e-01, -2.09257504e+00,
            8.12288670e-01,  1.18754540e+00,  1.02880847e+00,  2.09006018e+00,
           -2.65374169e-01,  2.60460392e-01, -1.29239184e+00,  2.97213013e-01,
            5.54126915e-01, -1.07758083e+00, -4.24177778e-01, -1.09759532e+00,
            5.67577251e-01, -1.20873008e-01,  8.23912142e-01, -8.05800702e-01,
           -1.92109475e+00, -1.22895780e+00,  7.68528239e-01,  1.71249945e+00,
            8.24228574e-02, -7.61190964e-01, -2.72912507e-01,  1.83584542e+00,
            6.72885061e-01, -4.45754189e-01,  1.21704102e+00,  1.92145180e-01,
            1.51088234e-01,  1.59421281e+00, -1.33765095e+00, -4.85955977e-02,
            1.00634056e+00,  4.69892652e-01,  9.07220541e-01,  4.32247353e-01,
           -1.66620371e-01, -8.53670669e-01, -1.22617747e+00,  5.99291659e-01,
           -2.66835590e+00, -4.58305790e-01, -1.12783883e+00, -3.18829542e-01,
            2.78953799e-01, -1.13456906e+00, -2.07971462e-01,  1.22118521e+00,
            4.80748014e-01, -5.01151925e-01, -1.31616443e+00, -6.79367555e-01,
            3.55068652e-01, -2.54682120e-01,  8.84540604e-01, -4.82088103e-01,
           -3.79796475e-01, -6.94715172e-01,  5.42787585e-01, -6.15518676e-01,
            1.33571333e+00, -7.53093053e-01,  1.59464820e-01, -3.42610338e-01,
            5.80974799e-01,  3.28274980e-01, -9.88035155e-01, -1.46675025e+00,
           -9.37054031e-01,  3.38921708e-01, -2.44935891e-01, -7.89146451e-01,
           -3.06348465e-01,  1.46735156e-01,  6.25030298e-01, -5.66648854e-01,
           -2.57801535e-01, -1.01865561e+00, -3.74768680e-01,  2.75270929e+00,
            4.58425841e-01, -3.96170878e-01, -6.38005380e-02, -1.21525840e+00,
           -1.23838271e+00,  2.90287988e-01,  1.14194480e+00,  5.68948251e-02,
           -1.85129600e+00, -2.51033526e-01,  8.25679104e-01, -3.46746831e-01,
            1.90964561e+00,  4.93143689e-01,  5.48691564e-01, -7.70165399e-02,
           -4.21678149e-03, -1.85590098e-01, -5.48877535e-03, -3.10558380e-01,
            2.41220649e+00,  1.50034552e+00,  9.79070079e-01, -5.34708082e-02,
           -2.73486319e-01,  9.03213412e-01,  2.15679497e-01, -1.29718753e+00,
           -9.65043353e-01, -7.79312710e-02, -1.27718813e+00, -1.24837109e+00,
           -3.37617288e-01, -8.73219558e-01, -7.07524465e-01,  4.15596111e-01,
            4.89763712e-01,  4.03658602e-01,  1.09086422e+00, -1.10907954e+00,
            9.10939858e-01,  1.31737298e+00, -5.76893677e-01,  1.20213536e-01,
            1.44081529e+00,  4.15123562e-01, -2.39042389e-01, -5.19599319e-01,
            4.94312960e-01, -2.35442242e+00,  8.63153490e-01, -1.32585306e-02,
            1.33631061e+00, -1.98589487e+00, -9.39927021e-01, -6.45488712e-01,
           -4.46334196e-01, -3.96770210e-01, -1.09963469e+00,  7.20233038e-01,
           -4.09746689e-01,  4.19550541e-01, -9.27710948e-01,  1.86459189e-02,
            4.83423364e-01,  5.42658361e-01,  6.08198560e-02,  2.26481384e+00,
            1.03788850e+00, -1.18080652e+00,  3.56716052e-01, -2.72060736e-01,
           -4.80617606e-01,  1.37980882e+00,  2.28088697e+00,  1.55715620e+00,
            6.30819552e-01,  1.26978350e+00, -5.60972123e-02, -1.65819606e+00,
           -8.47557228e-02, -7.58907663e-01, -1.93866968e+00,  8.07162652e-01,
           -1.72856201e+00, -7.77741973e-01, -1.90586911e-01, -2.79493074e-01,
           -2.41584752e-01, -4.80816958e-01,  5.04670916e-01,  6.47506095e-02,
           -3.17709797e-01,  5.45082790e-01, -9.11238672e-01,  1.03883342e+00,
           -4.06591132e-02,  1.79114512e+00, -5.99343444e-01,  3.67208439e-01,
           -1.36955178e+00, -1.68002211e+00,  8.11495174e-01,  9.69447164e-01])

    ipdb>  np.sqrt(ortho[:, 0] ** 2)

    array([0.98429131, 1.41339722, 1.2827928 , 0.01663013, 0.09914473,
           0.81230053, 0.95214137, 0.27462135, 0.10819478, 0.04855695,
           0.28010791, 1.28745879, 0.41824515, 0.81372469, 0.0220776 ])

    ipdb>  np.sqrt(sum(ortho[:, 0]) ** 2)

    0.21030968135506517

    ipdb>  np.sqrt(sum(ortho[0]) ** 2)

    21.479860284119574

    ipdb>  ortho[0]

    array([ 9.84291314e-01, -2.67154787e+00, -2.47711178e-01, -1.02029636e+00,
           -1.65410677e-01, -3.89257516e-01, -1.43501747e-01, -1.36789983e-02,
            1.11712103e+00,  3.73601046e-01, -2.39889307e-01, -9.41772074e-01,
           -2.72733437e+00,  1.90295545e+00,  7.78069913e-01,  6.15355372e-01,
            5.02227079e-01,  5.95148220e-01, -6.74557003e-01, -1.67540061e+00,
           -3.88751246e-01,  7.25295361e-01, -1.56139492e+00, -1.74060487e+00,
            4.27186242e-01, -3.91127127e-01, -2.70114437e-01, -1.03652805e-01,
            1.28952992e-01, -3.57169044e-01, -1.07204561e+00,  2.08143021e+00,
            5.44215711e-01, -3.68947001e-01,  9.92032509e-01,  4.65406769e-01,
            1.11568972e+00, -8.01249721e-02, -1.39841164e+00,  1.15093542e-01,
            1.50016540e+00,  1.00177466e+00,  1.05624218e-02,  5.71041762e-01,
           -1.19683936e+00,  1.19632552e+00,  9.66264547e-01,  1.25781920e+00,
            5.39271060e-01,  2.59142814e-01,  2.25129475e-01, -4.19216543e-01,
           -3.26935662e-01, -3.17915774e-01,  1.23800377e+00,  9.23184946e-01,
           -8.37310826e-01, -5.03856664e-02, -9.87949256e-02, -2.99166803e-01,
            2.28218814e+00,  8.51137604e-01,  7.31947502e-01, -2.32982144e+00,
            1.23826454e-01, -7.78613026e-01, -5.45202830e-01,  3.37359817e-01,
           -2.17156123e+00, -6.10592540e-01, -3.94995003e-02,  1.43910523e+00,
           -3.31352625e-01,  2.40060782e+00,  1.13188660e-01,  1.38870499e-01,
           -4.55218021e-01, -1.01446149e+00, -4.35811268e-01, -1.64153446e-01,
            5.24502470e-01,  3.39243052e+00,  3.59381107e-01,  3.86981560e-01,
           -2.51232959e+00, -1.21331210e-03, -1.21410687e-01,  6.67710798e-01,
            1.14347721e+00, -5.36425690e-01,  2.17181972e-01, -5.59598350e-01,
            9.75168230e-01, -5.14634205e-01, -5.81507896e-01, -4.77028875e-01,
           -1.74804083e+00,  1.60496408e+00,  1.13275195e+00, -6.95774124e-02,
           -2.12944326e+00,  5.09929330e-01,  1.16085934e+00,  5.94504870e-01,
           -4.81873905e-01,  1.67054591e-01,  9.35608651e-01, -4.60667543e-01,
           -5.66701133e-01,  2.99783302e-01, -1.40663804e-01,  2.82561049e-01,
           -4.45756576e-01, -1.73019799e+00, -3.21366614e-01,  1.36956498e+00,
            1.66702501e+00,  1.83065173e+00, -1.22085263e-01, -1.36478960e+00,
            1.74504022e-01,  1.75264518e-01, -6.16512548e-01,  1.12353659e+00,
            9.32440990e-01, -2.16105604e-01, -3.33619008e-01, -5.72438865e-01,
            7.50679815e-01, -8.91832022e-01, -5.97205414e-01, -4.74942807e-01,
            9.28396568e-01,  9.56712412e-01, -1.27674585e+00,  7.83552164e-01,
           -8.07671750e-01,  7.06130720e-01,  3.67023768e-01, -5.43778749e-01,
           -3.31812462e-01,  8.65133324e-01, -1.77622194e-01,  5.91300284e-01,
            1.70962732e+00, -1.47078518e+00,  1.20761351e+00,  1.20885814e+00,
           -1.24029244e+00, -6.63599727e-01, -1.46291702e+00, -6.35502592e-01,
            6.84424472e-01, -1.40903283e+00,  1.42180200e-01, -5.71668361e-01,
           -1.85136293e+00, -4.30382095e-01,  1.47271723e-01,  2.23333212e-01,
           -6.89780287e-01,  2.19725763e+00,  2.63307518e+00,  1.97236353e+00,
            8.23191353e-01,  5.56538342e-02, -3.52996919e+00,  1.41044467e+00,
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            7.94458573e-01,  1.60956234e-01, -1.82155993e+00, -2.94798391e-01,
            1.42499421e+00,  6.40541912e-01, -8.58036622e-01,  5.01906348e-02,
            2.10341824e-01,  2.27829896e-01,  1.73965430e-01,  2.13025461e+00,
           -7.26232261e-01, -3.37701641e-01,  4.30992835e-01, -5.89017127e-01,
            9.95471147e-01, -3.28638941e-01, -1.91851506e-01, -1.00146658e+00,
           -5.77852962e-01, -6.98284365e-01,  1.61031679e-01,  4.45699637e-01,
           -1.21426278e+00,  1.16179650e+00, -1.16362782e+00,  2.95560041e-01,
           -1.77993501e+00, -6.23721615e-01,  1.19663374e+00, -5.12522002e-01,
           -6.79025785e-01,  4.73970927e-03,  2.36189587e-02, -1.09738120e+00,
           -1.81525789e+00,  1.04241712e+00, -2.93739211e-01,  8.83260632e-01,
           -1.60817534e-01, -7.66919955e-01, -4.38373244e-02, -3.50647359e-02,
            4.79043236e-01,  1.79482102e+00, -1.09083112e-01, -2.75494498e-01,
           -1.11406559e+00, -2.75079324e-01, -6.23134082e-01,  3.04734162e-01,
            1.28850973e+00, -2.83485108e-01, -1.69655630e+00,  9.79401597e-01,
           -3.12492901e-01,  2.94454403e-01, -1.26683958e+00,  4.46599200e-01,
           -2.93666269e-01,  1.58311654e+00,  9.87463975e-01, -1.10681478e+00,
            7.95284927e-01, -5.82788181e-01,  2.27962712e-01,  1.04370482e+00,
            8.23308661e-01, -1.51070222e+00, -2.18168940e-01, -9.81920417e-01,
           -1.45499130e+00, -3.92751781e-01,  1.25545699e-01, -9.90084376e-01,
            1.00116657e+00,  2.47838167e-01,  3.00519072e-01, -2.82124234e-01,
            4.17485530e-01,  3.18968275e-01,  2.96229723e-01, -4.17821942e-01,
           -1.45871631e+00,  1.62479699e-01, -8.56423605e-03,  3.96696834e-01,
            4.18949089e-01, -1.43310801e-01,  7.07014221e-01,  7.01616614e-01,
           -1.98153662e+00, -8.74845568e-01,  1.87622408e-01, -6.15385817e-02,
            2.76841035e-01,  1.66629000e-01,  1.79155535e-01, -9.03926259e-01,
           -7.24157936e-01,  7.93521710e-01, -1.64952810e+00, -1.70205072e+00,
            2.05825595e-01,  7.40382379e-01, -5.37714462e-01, -1.52694761e+00,
            6.32964258e-01,  4.14327408e-01, -3.11672566e-01,  2.78290256e-01,
            5.39156542e-01,  2.83981209e-01,  9.59175834e-02, -1.42824551e+00,
           -5.39643678e-01, -8.76655697e-01, -1.03197996e+00, -8.68542043e-01,
           -1.12500399e+00,  1.90393480e+00,  2.65147680e+00, -5.59393893e-01,
            8.56445023e-02,  7.26434508e-01, -2.72738158e-01,  9.95835144e-01,
           -1.67793063e-01, -1.77687724e-01, -8.06109704e-01, -1.24820053e+00,
           -6.33703578e-01,  2.24870569e+00,  1.29512088e+00, -9.11925437e-01,
           -5.74153672e-01,  1.63023433e+00, -7.19184233e-01,  1.90290227e-01,
            4.22116506e-01, -9.23919995e-01,  1.11689852e+00,  1.33079877e+00,
           -4.92987846e-01,  6.88298610e-01, -1.03686606e+00, -1.12047763e+00,
           -8.23447240e-01,  3.43245350e-01, -8.78207308e-01,  1.61951865e+00,
           -1.97437355e+00,  1.75049893e-01,  1.88049874e-01,  8.59878182e-01,
           -1.05445976e+00, -1.22640363e+00, -7.78134507e-01, -3.76029154e-01,
            1.99887836e-01,  2.77795070e-02, -1.65239496e+00, -1.11167128e+00,
            1.97934406e+00, -7.14254156e-01,  1.55912743e+00,  1.57221889e+00,
           -9.83464734e-02,  5.78516832e-01,  1.82554545e+00,  2.48215884e-01,
           -7.68486404e-01, -7.53570547e-01,  1.59776693e+00, -1.62883237e-01,
           -1.06246786e+00, -3.67058822e-01,  1.04559196e+00, -2.89745964e-01,
           -2.51721276e+00,  1.22605302e+00,  1.48575112e-01, -2.09257504e+00,
            8.12288670e-01,  1.18754540e+00,  1.02880847e+00,  2.09006018e+00,
           -2.65374169e-01,  2.60460392e-01, -1.29239184e+00,  2.97213013e-01,
            5.54126915e-01, -1.07758083e+00, -4.24177778e-01, -1.09759532e+00,
            5.67577251e-01, -1.20873008e-01,  8.23912142e-01, -8.05800702e-01,
           -1.92109475e+00, -1.22895780e+00,  7.68528239e-01,  1.71249945e+00,
            8.24228574e-02, -7.61190964e-01, -2.72912507e-01,  1.83584542e+00,
            6.72885061e-01, -4.45754189e-01,  1.21704102e+00,  1.92145180e-01,
            1.51088234e-01,  1.59421281e+00, -1.33765095e+00, -4.85955977e-02,
            1.00634056e+00,  4.69892652e-01,  9.07220541e-01,  4.32247353e-01,
           -1.66620371e-01, -8.53670669e-01, -1.22617747e+00,  5.99291659e-01,
           -2.66835590e+00, -4.58305790e-01, -1.12783883e+00, -3.18829542e-01,
            2.78953799e-01, -1.13456906e+00, -2.07971462e-01,  1.22118521e+00,
            4.80748014e-01, -5.01151925e-01, -1.31616443e+00, -6.79367555e-01,
            3.55068652e-01, -2.54682120e-01,  8.84540604e-01, -4.82088103e-01,
           -3.79796475e-01, -6.94715172e-01,  5.42787585e-01, -6.15518676e-01,
            1.33571333e+00, -7.53093053e-01,  1.59464820e-01, -3.42610338e-01,
            5.80974799e-01,  3.28274980e-01, -9.88035155e-01, -1.46675025e+00,
           -9.37054031e-01,  3.38921708e-01, -2.44935891e-01, -7.89146451e-01,
           -3.06348465e-01,  1.46735156e-01,  6.25030298e-01, -5.66648854e-01,
           -2.57801535e-01, -1.01865561e+00, -3.74768680e-01,  2.75270929e+00,
            4.58425841e-01, -3.96170878e-01, -6.38005380e-02, -1.21525840e+00,
           -1.23838271e+00,  2.90287988e-01,  1.14194480e+00,  5.68948251e-02,
           -1.85129600e+00, -2.51033526e-01,  8.25679104e-01, -3.46746831e-01,
            1.90964561e+00,  4.93143689e-01,  5.48691564e-01, -7.70165399e-02,
           -4.21678149e-03, -1.85590098e-01, -5.48877535e-03, -3.10558380e-01,
            2.41220649e+00,  1.50034552e+00,  9.79070079e-01, -5.34708082e-02,
           -2.73486319e-01,  9.03213412e-01,  2.15679497e-01, -1.29718753e+00,
           -9.65043353e-01, -7.79312710e-02, -1.27718813e+00, -1.24837109e+00,
           -3.37617288e-01, -8.73219558e-01, -7.07524465e-01,  4.15596111e-01,
            4.89763712e-01,  4.03658602e-01,  1.09086422e+00, -1.10907954e+00,
            9.10939858e-01,  1.31737298e+00, -5.76893677e-01,  1.20213536e-01,
            1.44081529e+00,  4.15123562e-01, -2.39042389e-01, -5.19599319e-01,
            4.94312960e-01, -2.35442242e+00,  8.63153490e-01, -1.32585306e-02,
            1.33631061e+00, -1.98589487e+00, -9.39927021e-01, -6.45488712e-01,
           -4.46334196e-01, -3.96770210e-01, -1.09963469e+00,  7.20233038e-01,
           -4.09746689e-01,  4.19550541e-01, -9.27710948e-01,  1.86459189e-02,
            4.83423364e-01,  5.42658361e-01,  6.08198560e-02,  2.26481384e+00,
            1.03788850e+00, -1.18080652e+00,  3.56716052e-01, -2.72060736e-01,
           -4.80617606e-01,  1.37980882e+00,  2.28088697e+00,  1.55715620e+00,
            6.30819552e-01,  1.26978350e+00, -5.60972123e-02, -1.65819606e+00,
           -8.47557228e-02, -7.58907663e-01, -1.93866968e+00,  8.07162652e-01,
           -1.72856201e+00, -7.77741973e-01, -1.90586911e-01, -2.79493074e-01,
           -2.41584752e-01, -4.80816958e-01,  5.04670916e-01,  6.47506095e-02,
           -3.17709797e-01,  5.45082790e-01, -9.11238672e-01,  1.03883342e+00,
           -4.06591132e-02,  1.79114512e+00, -5.99343444e-01,  3.67208439e-01,
           -1.36955178e+00, -1.68002211e+00,  8.11495174e-01,  9.69447164e-01])

    ipdb>  np.dot(ortho[0], ortho[1])

    1.1102230246251565e-15

```python
fig, ax = plt.subplots(nrows=1, ncols=2, figsize=(4/3 * 2 * 5, 5))

plt.suptitle(r"'Overlap' vs interpolation parameter $\alpha$")
for idx, key in enumerate(["wwT", "wTw"]):
    ax[idx].set_title(key)
    ax[idx].errorbar(alphas, means[key], yerr=stds[key], label=key)
    ax[idx].set_xlabel(r"$\alpha$")
    ax[idx].set_ylabel("overlap")
plt.tight_layout(rect=[0,0,1,.95])
plt.show()
```

Note that orthonormalising the weights of the first teacher (interpreted as vectors in K dimensions) increases the fluctuations in $W^\top W$, but these fluctuations are in any case much larger than those in $WW^\top$.