latte.metrics.keras.interpolatability

Module Contents

Classes

Smoothness	Calculate latent smoothness.
Monotonicity	Calculate latent monotonicity.

class Smoothness(reg_dim=None, liad_mode='forward', max_mode='lehmer', ptp_mode='naive', reduce_mode='attribute', clamp=False, p=2.0)

Bases: latte.metrics.keras.wrapper.KerasMetricWrapper

Calculate latent smoothness.

Smoothness is a measure of how smoothly an attribute changes with respect to a change in the regularizing latent dimension. Smoothness of a latent vector \(\mathbf{z}\) is based on the concept of second-order derivative, and is given by

\[\operatorname{Smoothness}_{i,d}(\mathbf{z};\delta) = 1-\dfrac{\mathcal{C}_{k\in\mathfrak{K}}[\mathcal{D}_{i,d}^{(2)}(\mathbf{z} + k\delta\mathbf{e}_d;\delta )]}{\delta^{-1}\mathcal{R}_{k\in\mathfrak{K}}[\mathcal{D}_{i,d}^{(1)}(\mathbf{z} + k\delta\mathbf{e}_d;\delta )]},\]

where \(\mathcal{D}_{i,d}^{(n)}(z; \delta)\) is the \(n\) th order latent-induced attribute difference (LIAD) as defined below, \(\mathbf{e}_d\) is the \(d\) th elementary vector, \(\mathcal{C}_{k\in\mathfrak{K}}[\cdot]\) is the Lehmer mean (with p=2 by default) of its arguments over values of \(k\in\mathfrak{K}\), and \(\mathcal{R}_{k\in\mathfrak{K}}[\cdot]\) is the range of its arguments over values of \(k\in\mathfrak{K}\) (controlled by ptp_mode), and \(\mathfrak{K}\) is the set of interpolating points (controlled by z) used during evaluation.

The first-order LIAD is defined by

\[\mathcal{D}_{i, d}(\mathbf{z}; \delta) = \dfrac{\mathcal{A}_i(\mathbf{z}+\delta \mathbf{e}_d) - \mathcal{A}_i(\mathbf{z})}{\delta}\]

where \(\mathcal{A}_i(\cdot)\) is the measurement of attribute \(a_i\) from a sample generated from its latent vector argument, \(d\) is the latent dimension regularizing \(a_i\), \(\delta>0\) is the latent step size.

Higher-order LIADs are defined by

\[\mathcal{D}^{(n)}_{i, d}(\mathbf{z}; \delta) =\dfrac{{\mathcal{D}^{(n-1)}_i(\mathbf{z}+\delta \mathbf{e}_d) - \mathcal{D}^{(n-1)}_i(\mathbf{z})}}{\delta}.\]

Parameters:

• reg_dim (Optional[List], optional) – regularized dimensions, by default None Attribute a[:, i] is regularized by z[:, reg_dim[i]]. If None, a[:, i] is assumed to be regularized by z[:, i].

• liad_mode (str, optional) – options for calculating LIAD, by default “forward”. Only “forward” is currently supported.

• max_mode (str, optional) – options for calculating array maximum of 2nd order LIAD, by default “lehmer”. Must be one of {“lehmer”, “naive”}. If “lehmer”, the maximum is calculated using the Lehmer mean with power p. If “naive”, the maximum is calculated using the naive array maximum.

• ptp_mode (str, optional) – options for calculating range of 1st order LIAD for normalization, by default “naive”. Must be either “naive” or a float value in (0.0, 1.0]. If “naive”, the range is calculated using the naive peak-to-peak range. If float, the range is taken to be the range between quantile 0.5-0.5*ptp_mode and quantile 0.5+0.5*ptp_mode.

• reduce_mode (str, optional) – options for reduction of the return array, by default “attribute”. Must be one of {“attribute”, “samples”, “all”, “none”}. If “all”, returns a scalar. If “attribute”, an average is taken along the sample axis and the return array is of shape (n_attributes,). If “samples”, an average is taken along the attribute axis and the return array is of shape (n_samples,). If “none”, returns a smoothness matrix of shape (n_samples, n_attributes,).

• clamp (bool, optional) – Whether to clamp smoothness to [0, 1], by default False

• p (float, optional) – Lehmer mean power, by default 2.0 (i.e., contraharmonic mean). Only used if max_mode == “lehmer”. Must be greater than 1.0.

References

[1]

11. 14. Watcharasupat, “Controllable Music: Supervised Learning of Disentangled Representations for Music Generation”, 2021.

update_state(z, a)

Update metric states. This function append the latent vectors and attributes to the internal state lists.

Parameters:

• z (tf.Tensor, (n_samples, n_interp) or (n_samples, n_features or n_attributes, n_interp)) – a batch of latent vectors

• a (tf.Tensor, (n_samples, n_interp) or (n_samples, n_attributes, n_interp)) – a batch of attribute(s)

result()

Compute metric values from the current state. The latent vectors and attributes in the internal states are concatenated along the sample dimension and passed to the metric function to obtain the metric values.

Returns:

Smoothness array. See reduce mode for return shape.

Return type:

tf.Tensor

class Monotonicity(reg_dim=None, liad_mode='forward', reduce_mode='attribute', liad_thresh=0.001, degenerate_val=np.nan, nanmean=True)

Bases: latte.metrics.keras.wrapper.KerasMetricWrapper

Calculate latent monotonicity.

Monotonicity is a measure of how monotonic an attribute changes with respect to a change in the regularizing dimension. Monotonicity of a latent vector \(\mathbf{z}\) is given by

\[\operatorname{Monotonicity}_{i,d}(\mathbf{z};\delta,\epsilon) = \dfrac{\sum_{k\in\mathfrak{K}}I_k\cdot \operatorname{sgn}(\mathcal{D}_{i,d}(\mathbf{z}+k\delta\mathbf{e}_d;\delta))}{\sum_{k\in\mathfrak{K}}I_k},\]

where \(\mathcal{D}_{i,d}(z; \delta)\) is the first-order latent-induced attribute difference (LIAD) as defined below, \(I_k = \mathbb{I}[|\mathcal{D}_{i,d}(\mathbf{z}+k\delta\mathbf{e}_d;\delta)| > \epsilon] \in \{0,1\}\), \(\mathbb{I}[\cdot]\) is the Iverson bracket operator, \(\epsilon > 0\) is a noise threshold for ignoring near-zero attribute changes, and \(\mathfrak{K}\) is the set of interpolating points (controlled by z) used during evaluation.

The first-order LIAD is defined by

\[\mathcal{D}_{i, d}(\mathbf{z}; \delta) = \dfrac{\mathcal{A}_i(\mathbf{z}+\delta \mathbf{e}_d) - \mathcal{A}_i(\mathbf{z})}{\delta}\]

where \(\mathcal{A}_i(\cdot)\) is the measurement of attribute \(a_i\) from a sample generated from its latent vector argument, \(d\) is the latent dimension regularizing \(a_i\), \(\delta>0\) is the latent step size.

Parameters:

• reg_dim (Optional[List], optional) – regularized dimensions, by default None Attribute a[:, i] is regularized by z[:, reg_dim[i]]. If None, a[:, i] is assumed to be regularized by z[:, i].

• liad_mode (str, optional) – options for calculating LIAD, by default “forward”. Only “forward” is currently supported.

• reduce_mode (str, optional) – options for reduction of the return array, by default “attribute”. Must be one of {“attribute”, “samples”, “all”, “none”}. If “all”, returns a scalar. If “attribute”, an average is taken along the sample axis and the return array is of shape (n_attributes,). If “samples”, an average is taken along the attribute axis and the return array is of shape (n_samples,). If “none”, returns a smoothness matrix of shape (n_samples, n_attributes,).

• liad_thresh (float, optional) – threshold for ignoring noisy 1st order LIAD, by default 1e-3

• degenerate_val (float, optional) – fill value for samples with all noisy LIAD (i.e., absolute value below liad_thresh), by default np.nan. Another possible option is to set this to 0.0.

• nanmean (bool, optional) – whether to ignore the NaN values in calculating the return array, by default True. Ignored if reduce_mode is “none”. If all LIAD in an axis are NaNs, the return array in that axis is filled with NaNs.

References

[1]

11. 14. Watcharasupat, “Controllable Music: Supervised Learning of Disentangled Representations for Music Generation”, 2021.

update_state(z, a)

Update metric states. This function append the latent vectors and attributes to the internal state lists.

Parameters:

• z (tf.Tensor, (n_samples, n_interp) or (n_samples, n_features or n_attributes, n_interp)) – a batch of latent vectors

• a (tf.Tensor, (n_samples, n_interp) or (n_samples, n_attributes, n_interp)) – a batch of attribute(s)

result()

Compute metric values from the current state. The latent vectors and attributes in the internal states are concatenated along the sample dimension and passed to the metric function to obtain the metric values.

Returns:

Monotonicity array. See reduce mode for return shape.

Return type:

tf.Tensor