text:
\section{Context} \label{sec:intro}

Many of nowadays spectral unmixing techniques rely on non-negative matrix decompositions. This concerns for example hyperspectral remote sensing (with applications in Earth observation, astronomy, chemistry, etc.) or audio signal processing. The spectral sample $\ve{v}_{n}$ (the spectrum of light observed at a given pixel $n$, or the audio spectrum in a given time frame $n$) is decomposed onto a dictionary $\ve{W}$ of elementary spectral templates, characteristic of pure materials or sound objects, such that $\ve{v}_n \approx \ve{W} \ve{h}_n$. The composition of sample $n$ can be inferred from the non-negative expansion coefficients $\ve{h}_n$. This paradigm has led to state-of-the-art results for various tasks (recognition, classification, denoising, separation) in the aforementioned areas, and in particular in music transcription, the central application of this paper.

In state-of-the-art music transcription systems, the spectrogram $\ve{V}$ (with columns $\ve{v}_n$) of a musical signal is decomposed onto a dictionary of pure notes (in so-called multi-pitch estimation) or chords. $\ve{V}$ typically consists of (power-)magnitude values of a regular short-time Fourier transform \citep{sma03}. It may also consists of an audio-specific spectral transform such as the Mel-frequency transform, like in \citep{vinc10}, or the Q-constant based transform, like in \citep{ieee_taslp_2011b}. The success of the transcription system depends of course on the adequacy of the time-frequency transform \& the dictionary to represent the data $\ve{V}$. In particular, the matrix $\ve{W}$ must be able to accurately represent a diversity of real notes. It may be trained with individual notes using annotated data \citep{Boulanger-Lewandowski2012}, have a parametric form \citep{Rigaud2013a} or be learnt from the data itself using a harmonic subspace constraint \citep{vinc10}.

One important challenge of such methods lies in their ability to cope with the variability of real notes. A simplistic dictionary model will assume that one note characterised by fundamental frequency $\nu_{0}$ (e.g., $\nu_{0} = 440$ Hz for note A$_{4}$) will be represented by a spectral template with non-zero coefficients placed at $\nu_{0}$ and at its multiples (the {\em harmonic frequencies}). In reality, many instruments, such as the piano, produce musical notes with either slight frequency misalignments (so-called {\em inharmonicities}) with respect to the theoretical values of the fundamental and harmonic frequencies, or amplitude variations at the harmonic frequencies with respect to recording conditions or played instrument (variations of {\em timbre}). Handling these variabilities by increasing the dictionary with more templates is typically unrealistic and adaptive dictionaries have been considered in \citep{vinc10,Rigaud2013a}. In these papers, the spectral shape of the columns of $\ve{W}$ is adjusted to the data at hand, using specific time-invariant semi-parametric models. However, the note realisations may vary in time, something which is not handled by these approaches. This work presents a new spectral unmixing method based on optimal transportation (OT) that is fully flexible and remedies the latter difficulties. Note that \cite{typke2004searching} have previously applied OT to notated music (e.g., score sheets) for search-by-query in databases while we address here music transcription from audio spectral data.

\section{A relevant baseline: PLCA} \label{sec:plca}

Before presenting our contributions, we start by introducing the PLCA method of \cite{Smaragdis2006} which is heavily used in audio signal processing. It is based on the Probabilistic Latent Semantic Analysis (PLSA) of \cite{Hofmann2001} (used in text retrieval) and is a particular form of non-negative matrix factorisation (NMF). Simplifying a bit, in PLCA the columns of $\ve{V}$ are normalised to sum to one. Each vector $\ve{v}_n$ is then treated as a discrete probability distribution of ``frequency quanta'' and is approximated as $\ve{V} \approx \ve{W} \ve{H}$. The matrices $\ve{W}$ and $\ve{H}$ are of size $M \times K$ and $K \times N$, respectively, and their columns are constrained to sum to one. As a result, the columns of the approximate $\hat{\ve{V}} = \ve{W} \ve{H}$ sum to one as well and each distribution vector $\ve{v}_n$ is as such approximated by the counterpart distribution  $\hat{\ve{v}}_{n}$ in $\hat{\ve{V}}$. Under the assumption that $\ve{W}$ is known, the approximation is found by solving the optimisation problem defined by
\bal{
\min_{\ve{H} \ge 0} D_{\text{KL}}(\ve{V}|\ve{W} \ve{H}) \quad \text{s.t} \quad \forall n, \| \ve{h}_n \|_{1} =1,
}
where $D_{\text{KL}}(\ve{v}|\hat{\ve{v}}) = \sum_{i} v_{i} \log (v_{i}/\hat{v}_{i})$ is the KL divergence between discrete distributions, and by extension $D_{\text{KL}}(\ve{V}|\hat{\ve{V}}) = \sum_n D_{\text{KL}}(\ve{v}_{n}| \hat{\ve{v}}_{n})$.

 An important characteristic of the KL divergence is its separability with respect to the entries of its arguments. It operates a frequency-wise comparison in the sense that, at every frame $n$, the spectral coefficient $v_{in}$ at frequency $i$ is compared to its counterpart $\hat{v}_{in}$, and the results of the comparisons are summed over $i$. In particular, a small displacement in the frequency support of one observation may disproportionally harm the divergence value. For example, if $\ve{v}_{n}$ is a pure note with fundamental frequency $\nu_{0}$, a small inharmonicity that shifts energy from $\nu_{0}$ to an adjacent frequency bin will unreasonably increase the divergence value, when $\ve{v}_{n}$ is compared with a purely harmonic spectral template with fundamental frequency $\nu_{0}$. As explained in Section~\ref{sec:intro} such local displacements of frequency energy are very common when dealing with real data. A measure of fit invariant to small perturbations of the frequency support would be desirable in such a setting, and this is precisely what OT can bring.

\section{Elements of optimal transportation} \label{sec:OT}

Given a discrete probability distribution $\ve{v}$ (a non-negative real-valued column vector of dimension $M$ and summing to one) and a {\em target} distribution $\hat{\ve{v}}$ (with same properties), OT computes a {\em transportation matrix} $\ve{T}$ belonging to the set $\Theta \stackrel{\mbox{\footnotesize def}}{=} \{ \ve{T} \in \mathbb{R}_{+}^{M \times M} | \forall i,j = 1,\ldots,N, \sum\nolimits_{j=1}^{M} t_{ij} = v_{i}, \sum\nolimits_{i=1}^{M} t_{ij} = \hat{v}_{j}  \}$. $\ve{T}$ establishes a bi-partite graph connecting the two distributions. In simple words, an amount (or, in typical OT parlance, a ``mass'') of every coefficient of vector $\ve{v}$ is transported to an entry of $\hat{\ve{v}}$. The sum of transported amounts to the $j^{th}$ entry of $\hat{\ve{v}}$ must equal $\hat{v}_{j}$. The value of $t_{ij}$ is the amount transported from the $i^{th}$ entry of $\ve{v}$ to the $j^{th}$ entry of $\hat{\ve{v}}$. In our particular setting, the vector $\ve{v}$ is a distribution of spectral energies $v_{1},\ldots,v_{M}$ at sampling frequencies $f_{1},\ldots,f_{M}$.

Without additional constraints, the problem of finding a non-negative matrix $\ve{T} \in \Theta$ has an infinite number of solutions. As such, OT takes into account the {\em cost} of transporting an amount from the $i^{th}$ entry of $\ve{v}$ to the $j^{th}$ entry of $\hat{\ve{v}}$, denoted $c_{ij}$ (a non-negative real-valued number). Endorsed with this cost function, OT involves solving the optimisation problem defined by
\bal{
\min_{\ve{T}} J(\ve{T}| \ve{v}, \hat{\ve{v}}, \ve{C}) = \sum\nolimits_{ij} c_{ij} t_{ij} \quad \text{s.t} \quad \ve{T} \in \Theta, \label{eqn:otmin}
}
where $\ve{C}$ is the non-negative square matrix of size $M$ with
elements $c_{ij}$. Eq.~\eqref{eqn:otmin} defines a convex linear
program. The value of the function $J(\ve{T}| \ve{v}, \hat{\ve{v}}, \ve{C})$
at its minimum is denoted $D_{\ve{C}}(\ve{v}|\hat{\ve{v}})$. When $\ve{C}$
is a symmetric matrix such that $c_{ij} = \| f_{i} - f_{j}
\|_{p}^{p}$, where we recall that $f_{i}$ and $f_{j}$ are the
frequencies in Hertz indexed by $i$ and $j$,
$D_{\ve{C}}(\ve{v}|\hat{\ve{v}})$ defines a metric (i.e., a symmetric
divergence that satisfies the triangle inequality) coined Wasserstein
distance or earth mover's distance \citep{rubner98,villani09}. In
other cases, in particular when the matrix $\ve{C}$ is not even
symmetric like in the next section, $D_{\ve{C}}(\ve{v}|\hat{\ve{v}})$ is not
a metric in general, but is still a valid measure of fit. For
generality, we will refer to it as the ``OT divergence''.

By construction, the OT divergence can explicitly embed a form of invariance to displacements of support, as defined by the transportation cost matrix $\ve{C}$. For example, in the spectral decomposition setting, the matrix with entries of the form $c_{ij} = (f_{i}-f_{j})^{2}$ will increasingly penalise frequency displacements as the distance between frequency bins increases. This precisely remedies the limitation of the separable KL divergence presented in Section~\ref{sec:plca}. As such, the next section addresses variants of spectral unmixing based on the Wasserstein distance.

\section{Optimal spectral transportation (OST)} \label{sec:ost}

\paragraph{Unmixing with OT.}
In light of the above discussion, a direct solution to the sensibility
of PLCA to small frequency displacements consists in replacing the KL
divergence with the OT divergence. This amounts to solving the
optimisation problem given by
\bal{ \label{eqn:OTunmix}
\min_{\ve{H} \ge 0} D_{\ve{C}}(\ve{V}|\ve{W} \ve{H}) \quad \text{s.t} \quad \forall n, \| \ve{h}_n \|_{1} =1,
}
where $D_{\ve{C}}(\ve{V}|\hat{\ve{V}}) = \sum_{n} D_{\ve{C}}(\ve{v}_{n}|\hat{\ve{v}}_{n})$, $\ve{W}$ is fixed and populated with pure note spectra and $\ve{C}$ penalises large displacements of frequency support. This approach is
a particular case of NMF with the Wasserstein distance, which has been
considered in a face recognition setting by \cite{Sandler2011}, with subsequent developments by \cite{Zen2014} and \cite{Rolet2016}. This approach is relevant to our spectral unmixing scenario but as will be discussed in Section~\ref{sec:optim} is on the downside computationally intensive. It also requires the columns of $\ve{W}$ to be set to realistic note templates, which is still constraining. The next two sections describes a computationally more friendly approach which additionally removes the difficulty of choosing $\ve{W}$ appropriately.

\paragraph{Harmonic-invariant transportation cost.}

In the approach above, the harmonic modelling is
conveyed by the dictionary $\ve{W}$ (consisting of comb-like pure note spectra) and the invariance to small frequency displacements is introduced via the matrix $\ve{C}$. In this section we
propose to model both harmonicity and local invariance through the
transportation cost matrix $\ve{C}$. Loosely speaking, we want to define a class of equivalence between musical spectra, that takes into account their inherent harmonic nature. As such, we essentially impose that a harmonic frequency (i.e., a close multiple of its fundamental) can be considered equivalent to its fundamental, the only target of multi-pitch estimation. As such, we assume that a mass at one frequency can be transported to a divisor frequency with no cost. In other words, a mass at frequency $f_{i}$ can be transported with no cost to  $f_{i}/2$, $f_{i}/3$, $f_{i}/4$, and so on until sampling resolution. One possible cost matrix that embeds this property is
\bal{
c_{ij}= \min_{q=1,\dots,q_{\text{max}}} (f_{i}- q f_{j})^2 + \epsilon \, \delta_{q\neq 1}, \label{eqn:harmloss}
}
where $q_{\text{max}}$ is the ceiling of $f_{i}/f_{j}$ and $\epsilon$ is a small value. The term $\epsilon \, \delta_{q\neq1}$ favours the discrimination of octaves. Indeed, it penalises the transportation of a note of fundamental frequency $2\nu_{0}$ or $\nu_{0}/2$ to the spectral template with fundamental frequency $\nu_{0}$, which would be costless without this additive term. Let us denote by $\ve{C}_{h}$ the transportation cost matrix defined by Eq.~\eqref{eqn:harmloss}. Fig.~\ref{fig:transploss} compares $\ve{C}_{h}$ to the more standard quadratic cost $\ve{C}_{2}$ defined by $c_{ij}=(f_{i}- f_{j})^{2}$. With the quadratic cost, only local displacements are permissible. In contrast, the harmonic-invariant cost additionally permits larger displacements to divisor frequencies, improving robustness to variations of timbre besides to inharmonicities.

  \begin{figure}[t]
  \begin{center}
   \begin{tabular}{cccc}
         \includegraphics[height=33mm]{nips_loss1} &
         \includegraphics[height=33mm]{nips_loss2} &
         \includegraphics[height=33mm]{nips_loss3} &
         \includegraphics[height=33mm]{nips_loss4} \\
   \end{tabular}
  \caption{Comparison of transportation cost matrices $\ve{C}_{2}$ and $\ve{C}_{h}$ (full matrices and selected columns).}
  \label{fig:transploss}
    \end{center}
  \end{figure}

\paragraph{Dictionary of Dirac vectors.}
Having designed an OT divergence that encodes inherent properties of musical signals, we still need to choose a dictionary $\ve{W}$ that will encode the fundamental frequencies of the notes to identify. Typically, these will consist of the physical frequencies of the 12 notes of the chromatic scale (from note A to note G, including half-tones), over several octaves. As mentioned in Section~\ref{sec:intro}, one possible strategy is to populate $\ve{W}$ with spectral note templates. However, as also discussed, the performance of the resulting unmixing method will be capped by the representativeness of the chosen set of templates.

A most welcome consequence of using the OT divergence built on the harmonic-insensitive cost matrix $\ve{C}_{h}$ is that we may use for $\ve{W}$ a mere set of Dirac vectors placed at the fundamental frequencies $\nu_{1},\ldots,\nu_{K}$ of the notes to identify and separate. Indeed, under the proposed setting, a real note spectra (composed of one fundamental and multiple harmonic frequencies) can be transported with no cost to its fundamental. Similarly, a spectral sample composed of several notes can be transported to mixture of Dirac vectors placed at their fundamental frequencies. This simply eliminates the problem of choosing a representative dictionary! This very appealing property is illustrated in Fig.~\ref{fig:divcomp}. Furthermore, the particularly simple structure of the dictionary leads to a very efficient unmixing algorithm, as explained in the next section. In the following, the unmixing method consisting of the combined use of the harmonic-invariant cost matrix $\ve{C}_{h}$ and of the dictionary of Dirac vectors will be coined ``optimal spectral transportation'' (OST).

At this level, we assume for simplicity that the set of $K$ fundamental frequencies $\{ \nu_{1},\ldots,\nu_{K} \}$ is contained in the set of sampled frequencies $\{ f_{1}, \ldots,f_{M} \}$. This means that $\ve{w}_{k}$ (the $k^{th}$ column of $\ve{W}$) is zero everywhere except at some entry $i$ such that $f_{i} = \nu_{k}$ where $w_{ik}=1$. This is typically not the case in practice, where the sampled frequencies are fixed by the sampling rate, of the form $f_{i} = 0.5 (i/T) f_{s}$, and where the fundamental frequencies $\nu_{k}$ are fixed by music theory. Our approach can actually deal with such a discrepancy and this will be explained later in Section~\ref{sec:optim}.

\begin{figure}[t]
\begin{minipage}{16pc}
\includegraphics[width=\linewidth]{toysig3}
\end{minipage}
\begin{minipage}{16pc}
\footnotesize
\begin{tabular}[t]{|c|c|c|c|c|}\hline
	 Measure of fit & $D_{\ell_{2}}$ & $D_{\text{KL}}$ & $D_{\ve{C}_2}$& $D_{\ve{C}_{h}}$\\\hline
	$D(\ve{v}_1 |\hat{\ve{v}})$ & 1.13 & 72.92 & 145.00 &  134.32 \\
	$D(\ve{v}_2 |\hat{\ve{v}})$ & 1.13 & 5.42 & 10.00 &  10.00 \\
	$D(\ve{v}_3 |\hat{\ve{v}})$ & 0.91 & 2.02 & 1042.67 &  1.00 \\
	\hline
\end{tabular}
\end{minipage}

    \caption{Three example spectra $\ve{v}_{n}$ compared to a given template $\hat{\ve{v}}$ (left) and computed divergences (right). The template is a mere Dirac vector placed at a particular frequency $\nu_{0}$. $D_{\ell_{2}}$ denotes the standard quadratic error $\| \ve{x} -\ve{y} \|_{2}^{2}$. By construction of $D_{\ve{C}_h}$, sample $\ve{v}_{3}$ which is harmonically related to the template returns a very good fit with the latter OT divergence. Note that it does not make sense to compare output values of different divergences; only the relative comparison of output values of the same divergence for different input samples is meaningful.}
    \label{fig:divcomp}
  \end{figure}

\section{Optimisation} \label{sec:optim}

\paragraph{OT unmixing with linear programming.}

We start by describing optimisation for the state-of-the-art OT unmixing problem described by Eq.~\eqref{eqn:OTunmix} and proposed by \cite{Sandler2011}. First, since the objective function is separable with respect to samples, the optimisation problem decouples with respect to the activation columns $\ve{h}_{n}$. Dropping the sample index $n$ and combining Eqs.~\eqref{eqn:otmin} and~\eqref{eqn:OTunmix}, optimisation thus reduces to solving for every sample a problem of the form
\begin{align}
  \min_{\ve{h}\geq 0,\mathbf{T}\geq 0} \quad \langle\mathbf{T},\ve{C}\rangle = \sum\nolimits_{ij} t_{ij} c_{ij} \quad \text{s.t.} \quad \mathbf{T}\mathbf{1}_M=\ve{v},\quad \mathbf{T}^\top\mathbf{1}_M=\ve{W}\ve{h},    \label{eqn:LP}
\end{align}
where $\mathbf{1}_M$ is a vector of dimension $M$ containing only ones and
$\langle\cdot,\cdot\rangle$ is the Frobenius inner product. Vectorising the variables $\ve{T}$ and $\ve{h}$ into a single vector of dimension $M^{2} + K$, problem~\eqref{eqn:LP} can be turned into a canonical linear program. Because of the large dimension of the variable (typically in the order of $10^{5}$), resolution can however be very demanding, as will be shown in experiments.

\paragraph{Optimisation for OST.}

We now assume that $\ve{W}$ is a set of Dirac vectors as explained at the end of Section~\ref{sec:ost}. We also assume that $K < M$, which is the usual scenario. Indeed, $K$ is typically in the order of a few tens, while $M$ is in the order of a few hundreds. In such a setting $\hat{\ve{v}} = \ve{W} \ve{h}$ contains by design at most $K$ non-zero coefficients, located at the entries such that $f_{i} = \nu_{k}$. We denote this set of frequency indices by $\mathcal{S}$. Hence, for $j \notin \mathcal{S}$, we have $\hat{v}_{j} =0$ and thus $\sum_{i} t_{ij} = 0$, by the second constraint of Eq.~\eqref{eqn:LP}. Additionally, by the non-negativity of $\ve{T}$ this also implies that $\ve{T}$ has only $K$ non-zero columns, indexed by $j \in \mathcal{S}$. Denoting by $\widetilde{\mathbf{T}}$ this subset of columns, and by $\widetilde{\ve{C}}$ the corresponding subset of columns of $\ve{C}$, problem~\eqref{eqn:LP} reduces to
\begin{align}
  \min_{\ve{h}\geq 0,\widetilde{\mathbf{T}}\geq 0} \, \langle\widetilde{\mathbf{T}},\widetilde{\ve{C}}\rangle \quad \text{s.t.} \quad \widetilde{\mathbf{T}}\mathbf{1}_K=\ve{v},\quad \widetilde{\mathbf{T}}^\top\mathbf{1}_M=\ve{h}. \label{eq:unmixdirac2}
\end{align}
This is an optimisation problem of significantly reduced dimension $(M+1)K$. Even more appealing, the problem has a simple closed-form solution. Indeed, the variable $\ve{h}$ has a virtual role in problem \eqref{eq:unmixdirac2}. It only appears in the second constraint, which {\it de facto} becomes a free constraint. Thus problem~\eqref{eq:unmixdirac2} can be solved with respect to $\widetilde{\mathbf{T}}$ regardless of $\ve{h}$, and $\ve{h}$ is then simply obtained by summing the columns of $\widetilde{\mathbf{T}}^{\top}$ at the solution. Now, the problem
\bal{
\min_{\widetilde{\mathbf{T}}\geq 0} \, \langle\widetilde{\mathbf{T}},\widetilde{\ve{C}}\rangle \quad \text{s.t.} \quad \widetilde{\mathbf{T}}\mathbf{1}_K=\ve{v}
}
decouples with respect to the rows $\underline{\tilde{t}}_{i}$ of $\widetilde{\mathbf{T}}$, and becomes, $\forall i = 1,\ldots, M$,
\bal{
\min_{\underline{\tilde{t}}_{i} \ge 0} \sum\nolimits_{k} \tilde{t}_{ik} \tilde{c}_{ik} \quad \text{s.t.} \ \sum\nolimits_{k} \tilde{t}_{ik} = v_{i}.
}
The solution is simply given by $\tilde{t}_{ik_{i}^{\star}} = v_{i}$ for
$k_{i}^{\star} = \mathop{\mathrm{arg\,min}}_{k} \{ \tilde{c}_{ik} \}$, and $\tilde{t}_{ik} =
0$ for $k \not= k_{i}^{\star}$. Introducing the labelling matrix $\ve{L}$ which is everywhere zero except for indices $(i,k_{i}^{\star})$ where it is equal to 1, the solution to OST is trivially given by $\hat{\ve{h}} = \ve{L}^{\top} \ve{v}$. Thus, under the specific assumption that $\ve{W}$ is a set of Dirac vectors, the challenging problem~\eqref{eqn:LP} has been reduced to an effortless assignment problem to solve for $\ve{T}$ and a simple sum to solve for
$\ve{h}$. Note that the algorithm is independent of the particular structure of $\ve{C}$. In the end, the complexity per frame of OST reduces to $\mathcal{O}(M)$, which starkly contrasts with the complexity of PLCA, in the order $\mathcal{O}(KM)$ {\em per iteration}.

In Section~\ref{sec:ost}, we assumed for simplicity that the set of fundamental frequencies $\{ \nu_{k} \}_{k}$ was contained in the set of sampled frequencies $\{ f_{i}\}_{i}$. As a matter of fact, this assumption can be trivially lifted in the proposed setting of OST. Indeed, we may construct the cost matrix $\widetilde{\ve{C}}$ (of dimensions $M \times K$) by replacing the target frequencies $f_{j}$ in Eq.~\eqref{eqn:harmloss} by the theoretical fundamental frequencies $\nu_{k}$. Namely, we may simply set the coefficients of $\widetilde{\ve{C}}$ to be $\widetilde{c}_{ik} = \min_{q} (f_{i}- q \nu_{k})^2 + \epsilon \, \delta_{q\neq 1},\label{eqn:harmloss2}$ in the implementation. Then, the matrix $\widetilde{\mathbf{T}}$ indicates how each sample $\ve{v}$ is transported to the Dirac vectors placed at fundamental frequencies $\{ \nu_{k} \}_{k}$, without the need for the actual Dirac vectors themselves, which elegantly solves the frequency sampling problem.

\paragraph{OST with entropic regularisation (OST$_{e}$).}

The procedure described above leads to a {\em winner-takes-all} transportation of all of $v_{i}$ to its cost-minimum target entry $k^{\star}_{i}$. We found it useful in practice to relax this hard assignment and distribute energies more evenly by using the entropic regularisation of \cite{Cuturi13}. It consists of penalising the fit $\langle\widetilde{\mathbf{T}},\widetilde{\ve{C}}\rangle$ in Eq.~\eqref{eq:unmixdirac2} with an additional term $ \Omega_{e}(\widetilde{\mathbf{T}}) = \sum_{ik} \tilde{t}_{ik}\log(\tilde{t}_{ik})$, weighted by the hyper-parameter $\lambda_{e}$. The negentropic term $ \Omega_{e}(\widetilde{\mathbf{T}}) $ promotes the transportation of $v_{i}$ to several entries, leading to a smoother estimate of $\widetilde{\mathbf{T}}$. As explained in the supplementary material, one can show that the negentropy-regularised problem is a Bregman projection \citep{benamou2015iterative} and has again a closed-form solution $\hat{\ve{h}} = \ve{L}_{e}^{\top} \ve{v}$ where $\ve{L}_{e}$ is the $M \times K$ matrix with coefficients $l_{ik} = \exp(-\tilde{c}_{ik}/\lambda_{e}) / \sum_{p} \exp(-\tilde{c}_{ip}/\lambda_{e}) $. Limiting cases $\lambda_{e} =0$ and $\lambda_{e} = \infty$ return the unregularised OST estimate and the maximum-entropy estimate $h_{k} = 1/K$, respectively. Because $\ve{L}_{e}$ becomes a full matrix, the complexity per frame of OST$_{e}$ becomes $\mathcal{O}(KM)$.

\paragraph{OST with group regularisation (OST$_{g}$).}

We have explained above that the transportation matrix $\ve{T}$ has a strong group structure in the sense that it contains by construction $M-K$ null columns, and that only the subset $\widetilde{\mathbf{T}}$  needs to be considered. Because a small number of the $K$ possible notes will be played at every time frame, the matrix $\widetilde{\mathbf{T}}$ will additionally have a significant number of null columns. This heavily suggests using group-sparse regularisation in the estimation of $\widetilde{\mathbf{T}}$. As such, we also consider problem~\eqref{eq:unmixdirac2} penalised by the additional term $\Omega_{g}(\widetilde{\mathbf{T}})=\sum_{k} \sqrt{\|\widetilde{\mathbf{t}}_k\|_1}$ which promotes group-sparsity at column level \citep{Huang2009}. Unlike OST or OST$_{e}$, OST$_{g}$ does not offer a closed-form solution. Following \cite{courty2014domain}, a majorisation-minimisation procedure based on the local linearisation of $\Omega_{g}(\widetilde{\mathbf{T}})$ can be employed and the details are given in the supplementary material. The resulting algorithm consists in iteratively applying unregularised OST, as of Eq.~\eqref{eq:unmixdirac2}, with the iteration-dependent transportation cost matrix $\widetilde{\ve{C}}^{(iter)} = \widetilde{\ve{C}} + \widetilde{\ve{R}}^{(iter)}$, where $\widetilde{\ve{R}}^{(iter)}$ is the $M \times K$ matrix with coefficients $\widetilde{r}_{ik}^{(iter)} = \frac{1}{2} \| \widetilde{\ve{t}}_{k}^{(iter)} \|_{1}^{-\frac{1}{2}}$. Note that the proposed group-regularisation of $\widetilde{\mathbf{T}}$ corresponds to a sparse regularisation of $\ve{h}$. This is because $h_{k} = \| \widetilde{\mathbf{t}}_{k} \|_{1}$ and thus, $\Omega_{g}(\widetilde{\mathbf{T}})=\sum_{k} \sqrt{h_{k}}$. Finally, note that OST$_{e}$ and OST$_{g}$ can be implemented simultaneously, leading to OST$_{e+g}$, by considering the optimisation of the doubly-penalised objective function $\langle\widetilde{\mathbf{T}},\widetilde{\ve{C}}\rangle + \lambda_{e} \, \Omega_{e}(\widetilde{\mathbf{T}}) + \lambda_{g} \, \Omega_{g}(\widetilde{\mathbf{T}})$, addressed in the supplementary material.

\section{Experiments} \label{sec:exp}

\paragraph{Toy experiments with simulated data.}

In this section we illustrate the robustness, the flexibility and the
efficiency of OST on two simulated examples. The top plots of
Fig.~\ref{fig:toyexample}  display a synthetic dictionary of 8
harmonic spectral templates, referred to as the ``harmonic
dictionary''. They have been generated as Gaussian kernels placed at a
fundamental frequency and its multiples, and using exponential
dampening of the amplitudes. As everywhere in the paper, the spectral
templates are normalised to sum to one. Note that the $8$th template
is the upper octave of the first one. We compare the unmixing
performance of five methods in two different scenarios. The five
methods are as follows. PLCA is the method described in
Section~\ref{sec:plca}, where the dictionary $\ve{W}$ is the harmonic
dictionary. Convergence is stopped when the relative difference of the
objective function between two iterations falls below $10^{-5}$ or the
number of iterations (per frame) exceeds 1000. OT$_{h}$ is the
unmixing method with the OT divergence, as in the first paragraph of Section~\ref{sec:ost}, using the harmonic transportation cost
matrix $\ve{C}_h$ and the harmonic dictionary. OST is like OT$_{h}$,
but using a dictionary of Dirac vectors (placed at the 8 fundamental
frequencies characterising the harmonic dictionary). OST$_{e}$,
OST$_{g}$ and OST$_{e+g}$ are the regularised variants of OST,
described at the end of Section~\ref{sec:ost}. The iterative procedure in the group-regularised variants is run for 10 iterations (per frame).

In the first experimental scenario, reported in Fig.~\ref{fig:toyexample} (a), the data sample is generated by mixing the 1st and 4th elements of the harmonic dictionary, but introducing a small shift of the true fundamental frequencies (with the shift being propagated to the harmonic frequencies). This mimics the effect of possible inharmonicities or of an ill-tuned instrument. The middle plot of Fig.~\ref{fig:toyexample} (a), displays the generated sample, together with the ``theoretical sample'', i.e., without the frequencies shift. This shows how a slight shift of the fundamental frequencies can greatly impact the overall spectral distribution. The bottom plot displays the true activation vector and the estimates returned by the five methods. The table reports the value of the (arbitrary) error measure $\| \hat{\ve{h}} - \ve{h}_{\text{true}} \|_{1}$ together with the run time (on an average desktop PC using a MATLAB implementation) for every method. The results show that group-regularised variants of OST lead to best performance with very light computational burden, and without using the true harmonic dictionary. In the second experimental scenario, reported in Fig.~\ref{fig:toyexample} (b), the data sample is generated by mixing the 1st and 6th elements of the harmonic dictionary, with the right fundamental and harmonic frequencies, but where the spectral amplitudes at the latters do not follow the exponential dampening of the template dictionary (variation of timbre). Here again the group-regularised variants of OST outperforms the state-of-the-art approaches, both in accuracy and run time.

\begin{figure}[t]
\begin{minipage}{0.5\linewidth}
\centering \small
(a) Unmixing with shifted fundamental frequencies
\smallskip

\includegraphics[width=1\linewidth]{audio_example.png}

\setlength{\tabcolsep}{4pt}
\vspace*{2mm}
{\scriptsize
\begin{tabular}{|l|c|c|c|c|c|c|}\hline
Method & PLCA  & OT$_{h}$ & OST &OST$_{g}$&OST$_e$ &OST$_{e+g}$\\\hline
$\ell_1$ error &0.900  &     0.340 &   0.534  &  0.021& 0.660& 0.015\\
Time (s) &    0.057 &     6.541  & 0.006  &  0.007  &  0.007 &   0.013\\\hline
\end{tabular}
}
\end{minipage}
\begin{minipage}{0.5\linewidth}
\centering \small
(b) Unmixing with wrong harmonic amplitudes
\smallskip

\includegraphics[width=1\linewidth]{audio_example2.png}

\setlength{\tabcolsep}{4pt}
\vspace*{2mm}
{\scriptsize
\begin{tabular}{|l|c|c|c|c|c|c|}\hline
Method & PLCA  & OT$_{h}$ & OST &OST$_{g}$&OST$_e$ &OST$_{e+g}$\\\hline
$\ell_1$ error &0.791  &  0.430 &   0.971 &   0.045  &  0.911 & 0.048\\
Time (s) &    0.019 &   6.529 &   0.006  &  0.006  &  0.005 & 0.010\\\hline
\end{tabular}
}
\end{minipage}
\caption{Unmixing under model misspecification. See text for details.}
  \label{fig:toyexample}
\end{figure}

\paragraph{Transcription of real musical data.}

We consider in this section the transcription of a selection of real piano recordings, obtained from the MAPS dataset \citep{Emiya2010}. The data comes with a ground-truth binary ``piano-roll'' which indicates the active notes at every time. The note fundamental frequencies are given in MIDI, a standard musical integer-valued frequency scale that matches the keys of a piano, with 12 half-tones (i.e., piano keys) per octave. The spectrogram of each recording is computed with a Hann window of size $93$-ms and $50 \%$ overlap ($f_{s}$ = 44.1Hz). The columns (time frames) are then normalised to produce $\ve{V}$. Each recording is decomposed with PLCA, OST and OST$_{e}$, with $K=60$ notes (5 octaves). Half of the recording is used for validation of the hyper-parameters and the other half is used as test data. For PLCA, we validated 4 and 3 values of the width and amplitude dampening of the Gaussian kernels used to synthesise the dictionary. For OST, we set $\epsilon = q \epsilon_{0}$ in Eq.~\eqref{eqn:harmloss}, which was found to satisfactorily improve the discrimination of octaves increasingly with frequency, and validated 5 orders of magnitude of $\epsilon_{0}$. For OST$_{e}$, we additionally validated 4 orders of magnitude of $\lambda_{e}$. Each of the three methods returns an estimate of $\ve{H}$. The estimate is turned into a 0/1 piano-roll by only retaining the support of its $P_{n}$ maximum entries at every frame $n$, where $P_{n}$ is the ground-truth number of notes played in frame $n$. The estimated piano-roll is then numerically compared to its ground truth using the F-measure, a global recognition measure which accounts both for precision and recall and which is bounded between 0 (critically wrong) and 1 (perfect recognition). Our evaluation framework follows standard practice in music transcription evaluation, see for example \citep{Daniel2008}. As detailed in the supplementary material, it can be shown that OST$_{g}$ and OST$_{e+g}$ do not change the location of the maximum entries in the estimates of $\ve{H}$ returned by OST and OST$_{e}$, respectively, but only their amplitude. As such, they lead to the same F-measures than OST and OST$_{e}$, and we did not include them in the experiments of this section.

We first illustrate the complexity of real-data spectra in Fig.~\ref{fig:beats}, where the amplitudes of the first six partials (the components corresponding to the harmonic frequencies) of a single piano note are represented along time. Depending on the partial order $q$, the amplitude evolves with asynchronous beats and with various slopes. This behaviour is characteristic of piano sounds in which each note comes from the vibration of up to three coupled strings. As a consequence, the spectral envelope of such notes cannot be well modelled by a fixed amplitude pattern. Fig.~\ref{fig:beats} shows that, thanks to its flexibility, OST$_{e}$ can perfectly recover the true fundamental frequency (MIDI 50) while PLCA is prone to octave errors (confusions between MIDI 50 and MIDI 62). Then, Table \ref{tab:perfmusic} reports the F-measures returned by the three competing approaches on seven 15-s extracts of pieces from Chopin, Beethoven, Mussorgski and Mozart. For each of the three methods, we have also included a variant that incorporates a flat component in the dictionary that can account for noise or non-harmonic components. In PLCA, this merely consists in adding a constant vector $w_{f(K+1)} = 1/M$ to $\ve{W}$. In OST or OST$_{e}$ this consists in adding a constant column to $\widetilde{\ve{C}}$, whose amplitude has also been validated over 3 orders of magnitude. OST performs comparably or slightly inferiorly to PLCA but with an impressive gain in computational time ($\sim$3000$\times$ speedup). Best overall performance is obtained with OST$_{e}$+noise with an average $\sim$10\% performance gain over PLCA and $\sim$750$\times$ speedup.

A Python implementation of OST and real-time demonstrator are available at \url{https://github.com/rflamary/OST}

\begin{figure}[t]
\begin{minipage}{16pc}
\includegraphics[width=\textwidth,height=35mm]{Beats.png}
\end{minipage}
\begin{minipage}{16pc}
{\footnotesize
\begin{tabular}{c}
(a) Thresholded OST$_e$ transcription \\
\includegraphics[width=\textwidth]{note_OSTe}\\
(b) Thresholded PLCA transcription \\
\includegraphics[width=\textwidth]{note_PLCA}\\
{Time (s)}
\end{tabular}
}
\end{minipage}
\caption{First 6 partials and transcription of a single piano note (note D3, $\nu_{0}$ = 147 Hz, MIDI 50).}
\label{fig:beats}
\end{figure}

  \begin{table}[t]
    \caption{Recognition performance (F-measure values) and average computational unmixing times.}
    \label{tab:perfmusic}
    \centering \small
        \begin{tabular}{lcccccc}
\hline
MAPS dataset file IDs &  PLCA & PLCA+noise &    OST & OST+noise & OST$_e$ & OST$_e$+noise\\
\hline\hline
{\footnotesize \it chpn$\_$op25$\_$e4$\_$ENSTDkAm} & 0.679 & 0.671 & 0.566 & 0.564 & \textbf{0.695} & \textbf{0.695}\\
{\footnotesize \it mond$\_$2$\_$SptkBGAm} & 0.616 & \textbf{0.713} & 0.470 & 0.534 & 0.610 & 0.607\\
{\footnotesize \it mond$\_$2$\_$SptkBGCl} & 0.645 & 0.687 & 0.583 & 0.676 & 0.695 & \textbf{0.730}\\
{\footnotesize \it muss$\_$1$\_$ENSTDkAm} 4 & 0.613 & 0.478 & 0.513 & 0.550 & \textbf{0.671} & 0.667\\
{\footnotesize \it muss$\_$2$\_$AkPnCGdD} & 0.587 & 0.574 & 0.531 & 0.611 & 0.667 & \textbf{0.675}\\
{\footnotesize \it mz$\_$311$\_$1$\_$ENSTDkCl} & 0.561 & 0.593 & 0.580 & 0.628 & 0.625 & \textbf{0.665}\\
{\footnotesize \it mz$\_$311$\_$1$\_$StbgTGd2} & 0.663 & 0.617 & 0.701 & 0.718 & \textbf{0.747} & \textbf{0.747}\\
\hline\hline
Average & 0.624 & 0.619 & 0.563 & 0.612 & 0.673 & \textbf{0.684}\\
\hline \hline
Time (s)& 14.861 & 15.420 & \textbf{0.004} & \textbf{0.005} & 0.210 & 0.202\\
\hline
    \end{tabular}
  \end{table}

\section{Conclusions} \label{sec:conc}

In this paper we have introduced a new paradigm for spectral dictionary-based music transcription. As compared to state-of-the-art approaches, we have proposed a {\it holistic} measure of fit which is robust to local and harmonically-related displacements of frequency energies. It is based on a new form of transportation cost matrix that takes into account the inherent harmonic structure of musical signals. The proposed transportation cost matrix allows in turn to use a simplistic dictionary composed of Dirac vectors placed at the target fundamental frequencies, eliminating the problem of choosing a meaningful dictionary. Experimental results have shown the robustness and accuracy of the proposed approach, which strikingly does not come at the price of computational efficiency. Instead, the particular structure of the dictionary allows for a simple algorithm that is way faster than state-of-the-art NMF-like approaches. The proposed approach offers new foundations, with promising results and room for improvement. In particular, we believe exciting avenues of research concern the learning of $\ve{C}_{h}$ from examples and extensions to other areas such as in remote sensing, using application-specific forms of $\ve{C}$.

\paragraph{Acknowledgments.}
This work is supported in part by the
European Research Council (ERC) under the European Union's Horizon
2020 research \& innovation programme (project FACTORY) and by the French ANR under the JCJC programme (project MAD). Many thanks to Antony Schutz for generating \& providing some of the musical data.

\clearpage

redpajama_set_name: RedPajamaArXiv


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Brand Strategy, Values and Logos with Mandy Hall
On Episode 89 of the Cemoh Marketing Podcast Simon chats with Chiefs and Indians Founder, Mandy Hall.
Chiefs and Indians educates, advises and co-creates brand strategy and executions that rally people from the inside of an organisation, out.
You can contact Mandy Hall here https://www.linkedin.com/in/mandy-hall-09703615/
Simon Dell: Welcome to the very first Cemoh podcast – which we've done other podcasts in the past before, but this is the first one under the new brand. Welcome, Mandy Hall, who is back for her second time.
Mandy Hall: Yes. Thank you for having me again, and congratulations on Cemoh.
Simon Dell: Thank you, and congratulations to you because you are a new mother for the second time.
Mandy Hall: I am.
Simon Dell: And you had how much sleep last night?
Mandy Hall: Four hours sleep last night.
Simon Dell: A big poo explosion in the middle of the night?
Mandy Hall: A big poo explosion and a big coffee this morning.
Simon Dell: You poor, poor thing. I'm just getting out of all that now with an 18-month-old son.
Mandy Hall: Lucky you.
Simon Dell: I feel your pain. So, for those people that have listened to the podcast that you and I did in the past, they may know who you are and what you've done, but let's do a quick recap of where the hell you came from.
Mandy Hall: Well, I'm currently the director of a company called Chiefs and Indians and we do brand strategy projects, to keep it simple. I've spent the last 16 years in marketing and brand-related fields. I started at Virgin Blue, so pretty epic beginning in my career, and pretty much set me up for the rest, I believe, based on how I was in a brand role there.
And so, as the guardian of that brand, I've learned a lot in the power of what strong brand strategy and branding can bring to a company. And then from there, I worked in a range of marketing roles for some other big companies, smaller companies, government, and finished my corporate career before starting my own business at Compare the Market. So, again, finishing on a high note, prior to starting my own business which is where I am today.
Simon Dell: So, we won't talk about Compare the Market because we've talked about that last time. I don't want to rehash that. Before we talk about what we are going to talk about today, which is brand strategies, all of those kind of things, what made you want to go and work for yourself? At what point did you suddenly go, "Right, well, I've had enough of working for other people." What was the trigger?
Mandy Hall: I think it comes down to – there's a bit of frustration that I hold around branding and brand strategy, which is something that I've come across throughout my entire career. Firstly, a lot of people don't get it. They don't understand it. And you don't know what you don't know, right? There is no judgement there. People don't understand what it is. They think branding is a logo or branding is an advertising campaign, which it's definitely not.
You know, there's a real lack of brand strategy within organizations, so you need a good brand strategy before you can work out the best path to take in terms of branding. And so, I was very passionate about strategy and I sort of looked around. And there's a lot of creative advertising agencies that do brand strategy, but everything is wrapped up in coming up within an advertising idea or a creative idea.
And so, I sort of recognized there was a gap in the market for pure strategy. And certainly at a smaller level, there are a lot of other smaller businesses that are thriving that could really use some guidance and direction within their business but can't afford to bring in a full-time person.
I created Chiefs and Indians simply because I wanted to be able bring the 'Chiefs' and the 'Indians' to a range of business sizes and deliver brand strategies using global frameworks that had been proven to enhance commercial outcomes, removing any sort of fluff around brand, and it being the colouring-in department, and sort of bringing the substance, bringing the proof, bringing the insights that's going to drive a better commercial outcome.
Simon Dell: Right, so let's think about that in application to somebody who might be watching this, small to medium sized business. First question I've got is that you mentioned there's the people that think the brand is a logo. What, in simple terms, is a brand to you? What does that mean when you talk about brand?
Mandy Hall: Well, a brand is something that you really can't control. It's an imprint on the mind of your consumer or customer. It is a set of associations that come to mind about your business in the minds of the customer. Now when I say you can't control it, there are things that we do – and this is bringing it back to strategy – that you can try and influence their perceptions.
So, I always describe it as: Your brand strategy steers the ship. Your branding is the emblem on the sale, and your brand is the result of how well you did both. Just to clarify, branding isn't just the logo. The branding, the way we see it, is really the whole of your efforts in terms of a business. So, your branding comes from any sort of contact with the customer. It could be interactions, everything from your customer service, the people on the phone answering your calls, through to retailers in store, through to your marketing – you know, the most obvious stuff – your marketing, your website, and even your PI, the people that represent your company, it's all creating your brand.
Simon Dell: So if you – and this is what I see with a lot of small businesses – you've got really good customer service, they have really good products. They look after their customers really well. So, they are branding in that sense where people who are engaging with them on a day-to-day basis is really good?
Mandy Hall: Yes.
Simon Dell: But then when you look at their logos and the visual side of it, it looks like it was done in MS Paint in 1990. I see that a lot. Is that something you kind of…?
Mandy Hall: Yes. I see it go both ways because you have some companies that have the most beautiful branding, right? But their words are cheap, so they don't have the actions to follow it up. And then there are other companies like you have described who are creating a fantastic brand. From my point of view, product and service does trump any sort of advertising message. Because if you're not delivering on the advertising message, then it's worthless.
But it's about getting alignment between your visual, your verbal, and your service identities, your products even. I guess the first step there is coming up with a strategy so that there's complete clarity and razor-sharp focus across every department within a business. So, your brand strategy shouldn't sit with the marketing team. Yes, they might lead it, they might drive it, but it needs to be owned and implemented from the top-down.
Because what I've learned is when I created brand strategies in the past, and in my business, if you're not working with the CEO or the MD, and they haven't adopted it, then it's going to dissipate, it goes nowhere. Our proposition, what I am most passionate about, is creating brand strategies that are actionable – which goes against the fun and the bright lights, but I prefer to deliver something that is going to be actionable and results-orientated than something that becomes like a coffee table book.
Simon Dell: Where does the visual side of the brand then fit in to that kind of… Is that the last thing that you look at?
Mandy Hall: Yes. So basically, when we look at businesses, we don't necessarily go through a linear approach because some people have bits and pieces done. It's about bringing it together. When we do a brand strategy, it is broken up into two parts. The first is the strategy, which is we break it in to three sections which is impact, appeal, and character. So, that hasn't touched the branding yet. The character is very much just establishing the personality and the tone of voice, because you need that to then create the expression.
And then the branding is the second part, where you go, "Okay, this is our strategy. This is our vision. This is our values. This is how we want to position ourselves and this is the personality. Okay, so how do we then translate that into a creative expression?"
So we say strategy is internal, eyes only. Branding and expression is external. So that directs, but you've got the framework and the foundation, I should say, and then now it is creating that emotional, engaging, empathetic branding identity system that wins your customers over.
Simon Dell: Why is that all important? It sounds important, the way you describe it makes it sound very important, but what is the… You know? Again, going back to the example of that small business where they've got great customer service, they've got great product, but they've got the MS Paint logo. And then your one, which is they've got the great visual elements, but the rest of it seems to be a bit broken. Why bother? If the company is making and generating revenue and it is making its profit, what's the point?
Mandy Hall: Well, I guess it's how high do you want to strive. So, that sounds like an ideal scenario of the company is making revenue, making profit with their MS Paint logo. I mean, you don't just change shit because there's some sort of pragmatism that needs to be applied but I guess it's just a matter of getting all of your ducks in a row because in the end it's consistency and alignment that helps you become known.
That company, they might have all their ducks in a row, with the Paint logo and it may not be necessary. But if that logo represents their brand strategy or their brand positioning, then so be it. It doesn't matter if it is. I guess it is just a matter of making sure that – and it's particularly a problem as you grow. You can have this great brand and you have five people working for you, or ten people, or twenty people, but then you grow to two hundred people.
All of a sudden, documentation, processes, frameworks need to be crystal clear because there is so much more error that can come with more people. Essentially, when you do a brand strategy and build a brand right, by that I mean, there are companies around the world that have come up with the key elements that make a strong brand based on their research that has determined that brands that have the biggest commercial returns, act in these five ways. Well then, to me, it's just making sure that you've integrated those things into your brand consistently to then succeed longer term.
Simon Dell: I think what you've said there is about expanding the business. If you are going to do a small business and there is five of you, and that's all you've ever aspire to be, you don't want to sell the business, or you don't want to replicate it in another city or things like that. If your service levels are great, your proposition is great, and you have nothing written down, it doesn't really matter.
Mandy Hall: No. If those five people are clear, it's all about clarity. That's the whole point of this. Also, you might have a – pretty important too, if your business is introducing new products. So that's also where it can come undone if you are not clear about your customer and the types of needs, and wants, and desires and the products most suited to that. Because if you've got a proposition where you want to be the fastest, if you're a warehouse, or an Amazon or something, and you want to do everything in three clicks, that touches your website, that touches your products, that touches your warehouse. You see how there's all these things that – so you just want to make sure that you're keeping within that, or you can be all things to everybody, but that will absolutely will come undone eventually.
Simon Dell: I guess also from the benefit of having a strong brand, both with the values and also visually and all of those kind of things, is that itself becomes an asset. People will always talk about the value of the Coca-Cola logo. If you sold everything else from Coca-Cola, all the bottling plants, and all of those kind of things, the logo itself and the brand values that are epitomized by Coca-Cola is still worth billions of dollars. So, I guess that is the other thing, is you build a strong brand, and it creates more value within the business itself.
Mandy Hall: Absolutely. I was actually just looking at my social media the other day, and we did a post about why build a strong brand. And first thing, at the end of the day, the reason we try to build a brand is to spend less acquiring customers. Secondly, if you build a brand, you have potentially a good brand, less sensitivity to price. So, you could potentially charge a higher price. And lastly, you can sell more of your other stuff because your brand sits outside of, to your point, any one product or service. All of that just improves profits. Like, that's the whole point.
Simon Dell: People come back more frequently to buy from a brand they like and they trust.
Mandy Hall: Absolutely. But you can break their trust in a second, so that's the alignment, that's the consistency.
Simon Dell: I think about some of the brands that I would buy or have bought frequently. Apple is the global number one brand that when you talk about a brand that has built that kind of loyalty within its customer base over the years. People will come and buy an iPhone or a Mac, or whatever it is, without even really worrying about what the price is. There are other great brands as well that do that sort of thing, and then you see some that are polarising. You see the Tesla of this world where they've got the brand fan boys.
Mandy Hall: And that's okay. That's who they want. That's the beauty of certain niches. You don't need to be all things to everybody. I went to an event once, and they used the best quote, which was, "People don't buy products or services, they buy better versions of themselves." And I'm like, "That's so true!" That's what you're trying to create as a brand: This sort of emotional, higher connection. And it doesn't have to be aspirational. It can still be rational, but it's something that you can connect with and feel good about, to your point, unconsciously.
Simon Dell: Richard Branson is another person that's built really good brands throughout the world – you worked for Virgin Blue. And I'll never forget when he was trying to launch Virgin Cola in the U.S., that he drove a tank down, I think, Fifth Avenue, or something like that, like he's invading the heartland of America or New York with this new brand.
And immediately, just driving a tank down Fifth Avenue, there's your brand values. You are challenging the status quo, you're not afraid to get in a fight. You're having fun. You're doing something completely crazy. Their single act is the brand values epitomize that new product.
Mandy Hall: Exactly. And working at Virgin taught me a lot about brand values. They're not just words on a piece of paper. You've got to live and breathe them. And often, they can make you very distinctive as a brand and even differentiated. Sometimes, your positioning can be wrapped around parts of your brand values. Because your brand values is one piece of the puzzle, right? You've got to do these other things together for the greatest impact.
But what I loved about Virgin when they came into the market, and I started not long after – fun was one of their values. This is going back 18 years. That's when I first started at Virgin, which is really frightening. But back then, fun as a brand value – it was exactly that, and that was part of the proposition to the customer. This is the beauty of your values, and this is how values work two-fold. They're about creating a way of working, or some guiding principles for your staff and your employees, and you use those to attract people.
You get people excited about your brand. But then, the whole point, when you do brand values that are the same inside and out – and lots of businesses do different things with values. But the brand values that I'm talking about, the way we do it, is what they are internally, they are externally, and the value proposition to the customer externally. For example, fun was a value, and what we did at Virgin was the ads were super fun and super cheeky. Every time Richard Branson was in town, he threw a party for staff.
And things like on-board aircraft, they made jokes. You could never make jokes on board an aircraft. I'm not saying this is to everyone's taste. This is just showing that this is not words on paper. They had actions. They had customer service. Their advertising pushed the needle. And internally, we had parties. Richard Branson did outlandish things. Everything he did had this great, fun, PR-able moment.
I remember when I started carbon offsetting the flights. The PR team created this big green ball, and they had all these girls there. And I'm not saying that is appropriate today. But this was pushing the needle. And you know what? It created salience, which is a big part of building a brand: Being memorable, and having something that distinguishes you from others – and that did.
You watch where Virgin has come to today. They made a huge strategic change a while ago. I don't know what is going on behind closed doors, but from outside, it looks like they're replicating Qantas. I know for a fact that they dropped the fun value, because you can do that with your brand. You can grow out of what you used to be. They dropped their fun value – and you felt it. All of a sudden, they're a bit more premium, they've got business class.
But interestingly, it was the fun that really differentiated them. And now, I hear whispers about, "Oh, we want to be the old Virgin again." I'm not saying that it's true or this is just an outsider's point of view, but because I worked there so many years ago, I can see how things have changed, and it just shows you how you can change with your values.
Simon Dell: Let's just talk about those values for a second, because fun as a value is obviously – it's fairly obvious. You don't have to overthink that one. If you are out there in your business and going, "Right, we've been operating, we don't even know what our values are." How would you work those out? And two-part question, and can they just be a single word, fun, or should they be some sort of wanky sentence that feels a bit kind of…?
Because you read a lot of values and go – I've written them myself, I've written them for us. And I'll look at them and go, "God fuck, that's wanky."
Mandy Hall: Before I answer your question, I will say that it comes down to each individual. You are the CEO of your company, correct? You cannot have a wanky value. It will not come to life. I'm being real – it won't come to life. And so, first and foremost, it's not about… I sit with my clients and work through how to reach their values or how to articulate their values. And do you know what? I stand back. I'm the facilitator at best, because if I push what I believe and they don't believe it, nothing will happen and it's a waste of time.
And so, I often just start with – forget any sort of definitions of what it should be or what it shouldn't be. It's kind of like, what do you value? What are the behaviours? What do you care about? What are your non-negotiables in business? What don't you tolerate? And so, working through just some guiding principles, you can start simply by writing them on the board to get a feel for what they are.
Now, when I write values with my business copywriter, or when we're putting them together, I personally like articulating them in a way where you have a headline. You have two to three words. And I do this because anything decent needs to be repeatable. Because you want people to adopt it. Remember, I'm all about actions, and this is my learnings to get to here.
So, two to three words as a headline that sum up your value. I prefer of a verb so that it's more of a doing style. A few of my values are: awaken possibilities, take courageous action, keep it simple, being more human. See, they're more verbs for the team to really understand, but they also work externally.
Simon Dell: So how am I going to create a value around not being wanky?
Mandy Hall: You know what? I have been at a company where one of the values was "don't be a dick" or something, or maybe that was the Google one. We had something similar. Actually, I remember what it was now, but I can't repeat it because the visual used ended up becoming the President of America, but you get the gist.
You can have values like that, but you've got to explain them and you've got to flesh them out. Two to three words is a headline, and then I always think you have a little sentence or a paragraph that explains it. But what I do behind the scenes with my clients is, yes, you have short, sharp. You can put them up on a wall for employees to see and feel,, and you can recruit people on the basis of those things, because you want to emulate them externally. But necessary is to sit down and you go, "How do we do this"?
Simon Dell: "How do we make that brand come alive?"
Mandy Hall: Because then I'll think you'll find that the wanky-ness disappears. So, you're just looking at face-value. And look, some companies like single words. This is just our approach.
Simon Dell: There is no right answer.
Mandy Hall: Exactly. This is just how we do it. And all I will say is: Don't be generic. Don't put 'integrity' and 'innovation'. So what? That doesn't attract people. That doesn't attract customers. It's vanilla. You don't have to be wanky or creative, but just try and think about it slightly differently that makes it meaningful.
Simon Dell: And you don't have to get it right the first time.
Mandy Hall: No, no.
Simon Dell: These are evolutionary values that could change. I mean, you mentioned Virgin straightaway, but they're values that could change over the time of the course of the business' life cycle.
Mandy Hall: Yes. Chiefs and Indians has been around for three years, and we are about to go through a complete change. And my values have changed because we've learnt, we've understood. That wasn't quite us, but this is us. This is what resonates with my team, and this is what resonates with our customers.
Simon Dell: It's in one of the values that we've been talking about ourselves here, is the idea that the team here take responsibility for their own actions. So, one of the things that we like is people who work for us going – rather than keep asking us for permission to do things, to almost do something or fail, and then learn from that and go, "Well, this is what I did." And come to us and say, "This is what I did and it didn't work. Then I did this and it still didn't work, and then I did this and this is really working well."
And we're sitting there going, "That's great. Because if you keep having to ask us permission at each step of this, that takes up our time, that takes up your time, that slows the process down." It's that kind of whole move fast and break things.
Mandy Hall: Yeah, or ask for forgiveness, not for permission.
Simon Dell: Yeah, absolutely. But I think the "move fast and break things" was Steve Jobs, maybe? But that was his idea. It was just keep going, keep going, keep breaking things, keep smashing things, keep knocking things over. And eventually, you'll get to the solution much quicker than sitting there on your hands going, "I don't know whether I should do this or not" So, that's one of those values that we kind of then want to try and encapsulate into a single headline or a sentence.
Mandy Hall: Yeah, and that's brilliant and that resonates. Like, that's your business. This is the part of the proposition to the customer that you are on the cutting-edge of digital marketing or marketing solutions, because we are learning, and we are changing, and we are optimizing. So that in itself goes to show what you're delivering to the customer, but your expectation of your employees.
Simon Dell: The interesting thing is, now that I think about it, I lost a prospect on Tuesday this week because I explained that approach, probably not in that concise words that I've used with you, but I explained that approach to them. When they said to me, "Here's what we want to achieve with our product website and we want to achieve this amount of sales in this time frame."
And the question he said was, "Do you think you can do that?" And my answer at the bum was, "I haven't got a fucking clue." I've got no idea whether I could achieve that. I don't know your website. I don't know your products. I don't know your traffic. But I said, "Our job is to go back and start testing and trying, and start with a small amount of money and improve." Move fast, break things. It was funny like, the next email I got back, "I decided to go with someone else" who promised them that he could do this, and this, and this. And I'm like, "I'm glad I didn't get it!"
And I said to Matt, I said, "This one's going to be hard work." And it proved it. And just by being honest, I lost it and then sat back and went, "I'm glad I lost it."
Mandy Hall: I'm okay with that though, because I think part of your values – and when I talk about values, I'm not saying you get a client prospect, you do a proposal with your values, "Here's my values" kind of thing. It's not that way that I'm suggesting. But you set certain standards and approaches that enable you to accept and decline customers.
I often think about – the name of my business is Chiefs and Indians, right? Some people will think "that is absolutely ridiculous!" And do you know what? They are probably not the clients that I want. I use it as a bit of a filter. And that's okay. It doesn't make them bad or me bad. There's not that synergy, and I think it's okay to knock back clients because they're going to be disappointed. That guy would be disappointed with you, as you test, and learn, and optimize. And even though you'd probably double it in the end, he wouldn't have liked the process.
Simon Dell: Yeah, and he would've got frustrated too early on. It needs someone to say, "We're in this for twelve months" or whatever and we get go, right. All of the good shit is going to happen in the last three months when we get to that point but the first three months, six months… Probably not going to look great.
Mandy Hall: That's having standards put on you.
Simon Dell: We've obviously gone through a rebrand. One of the purposes of having this conversation: What do you use to trigger internally within a business, that they sit there and go, "We need to change our brand in the values space, as well as the strategies space, as well as the visual space." What do you see as normal triggers where that actually happens?
And I answer that in our case to say it was, we wanted to present a different product in the market. An evolution of our product, not a completely different product. And what we had as a brand was very much focused around Matt and I, the new brand that we wanted was we wanted to be very much focused on other people. So for us, it was saying, "We need to go from here to here, and it needs a new name." Et cetera. And we've managed to find a cool, five-lettered.com. But what are some of the other triggers that you use?
Mandy Hall: Well, that's probably a primary trigger. So, when people are repositioning, or changing their value proposition, and their old brand doesn't seem to fit anymore… Believe it or not, branding is a huge trigger. So, a lot of people come to me and they're like, "We've never invested in that brand," or "Our brand is tired." And you know, I'm not a graphic designer agency. And so straight away, I'm like, "Okay, so when you're going on an evolutionary process, or a revolutionary process, you need to understand why, what, and how first."
If you are after just a new logo, that's not me. A graphic design company can do that for them. Branding is often probably the number one reason people come to me, to find out how we change the brand. New competition, new marketplace, new products. So, I do work with people on product branding too, so not the corporate branding that we've been talking more about, but product branding. So, they're probably the core reasons.
Sometimes, I've had people come to me about marketing and then realize they don't have a business plan. And then I've ended up doing brand strategy, business plan, marketing strategy just because I'm probably not textbook brand strategy. There are certain things that are probably more from your business plan that I expect you to have in order for me and my team to develop a brand strategy for you. That's probably a big one. It's just once you start talking to people again, you don't know what you don't know.
Simon Dell: And often sometimes, I see the trigger is: yes, they do have a shitty logo and they go, "I don't really like our logo anymore. We need some brochures done." It's like putting the thread on a jumper, and all of a sudden, the whole thing unravels and they go, "Yes, so you got no values. You don't know what you stand for." And all of a sudden, you are sitting there doing a full brand strategy.
Mandy Hall: Exactly. So, they are often the triggers for…
Simon Dell: With us, actually, it's the website as well. They go… We doomed a website the other week, and we couldn't find a developer who even wanted to touch it. We're sitting there going, "Maybe it's time to think a new website. When was this built?" And they go like "8 years ago." And we go, "Hmm, yeah. You can probably do with a new website. It's not particularly mobile friendly et cetera."
And then they go like, "Alright, well while we are going to do our website, we want to do our logo and redo the branding, and all of those kind of things." And then all of a sudden, the Pandora's Box is open, and bang, there's all these things that you want to do.
The danger with us in the past, is we'll sit there and try and fix one bit. We gaffer tape over, "Yeah, we'll do a new website and use your old logo." And then you look at it again, "Yes, it doesn't really say what they do, it doesn't inspire me. It's not engaging." For most people, they've got to go right back to the start. "What do we actually stand for? What is this brand?" To your point, it's perhaps even further back than that. What's the point of this business? What are we here for?
Mandy Hall: Yeah, it's a really hard question for most people.
Simon Dell: I ask those questions so many times. The guys here joke with me about it because I always say to people, "Do you have a business or do you have a job? Have you just created yourself a job that pays you alright, and you get to have whatever days off that you like? But really, you've got a job." And unless you're going to think about selling it, or franchising it, or expanding it, or whatever it is, it's a job. And if it relies entirely on you or 80% of it, it's a job. It's not a business.
And that, to me, brand is kind of not really important. But if you want to translate your life from having a job to owning a business, you have to think about that, the brand strategy."
Mandy Hall: I saw a quote that was – and I might get this wrong because I'm a little bit foggy, "Start how you intend to finish." That's my new motto. Well, it's not as new as such, I probably already do that, but that's how I think about it. You never want to push this stuff on any businesses, because it can be a long-winded and expensive process. It's just about going, "Do you want a logo change? You can go to a graphic designer."
But a graphic designer is still going to say to you, "What do you stand for? Who are you? I want to get a feel for this brand so I'll know how to design for you." Or you are looking at more long-term, because a brand strategy isn't a marketing plan that is going turn a visitor around tomorrow. Although, it can do with some time. There's some result in branding. And certainly, direct marketing or performance marketing is enhanced by brand over the top.
I've learned that working for big brands, where the investment in brand is huge. But that's not feasible for many businesses nor necessary, but if you think about it as a percentage of your time.
Simon Dell: If you are a small business and can't afford the huge amount of dollars that it takes to hire somebody like yourself, what are some of the things that you could do yourself? Is there sort of a little exercise? Is it worth getting family and friends around the table?
Simon Dell: Not family and friends?
Mandy Hall: Well, everyone has a point of view. Everybody is a marketer. Don't cast it wide. If you do, cast it out to your customers, talk to them.
Simon Dell: So you're saying specifically don't get family and friends involved? Because I agree with you. Family and friends give you the worst advice.
Mandy Hall: Exactly, and they sort of derail you, and I start second guessing myself.
Simon Dell: If they don't like something just personally in their lives – I've seen it with logos, "I don't like green so I don't think that you should have it on your logo." Yeah, but we're an environmentally-friendly brand. You get completely derailed by family and friends.
Mandy Hall: Exactly. So, I wouldn't start there. I actually have a course to do this. I'm evolving my business model that's going to offer a brand strategy course, but this wasn't meant to be a sales pitch so I'm going to stop there.
Simon Dell: We'll put it in the notes or something.
Mandy Hall: Yeah. There's a course coming or there is one now but it's not currently live. But I would just suggest that you've got so much intelligence that people don't even realize. We start with insights. It's always about understanding first, trying to put your bias at the door and understand first. And what you can gather – I mean, I've worked with small businesses where I simply analysed two years of their business performance and was able to extrapolate segments based on their data capture.
And it wasn't advanced, and so, we were able to start creating segments. And then we were able to look at the average revenue per sale. We're just starting to piece together a picture. I know this isn't brand, but we are starting to piece together: Who are the valuable segments? Who are the most profitable? Who are the high-volume? And so you go, "Okay, these guys are your lovers. They love your service, okay? There is some strong segment there. I want to go and talk to them now and hear what they have to say about your brand."
When I do customer interviewers for clients, I try and let the customer do a lot of the talking to get an understanding, to actually just use some of their words. So, this isn't paying for qualitative research focus groups, because I'm a believer of having something as opposed to nothing, but this is just getting some insights by your customers, particularly the ones that love you – and then you can talk to maybe some that haven't loved you so much and what their issues were.
And then you're starting to piece together a little bit of a picture. If there's one question that I love asking customers in this industry, "Any other brand that comes to mind?" That's what you say. And straight away, you know who the front-runners are. Because you want to talk to at least 10-20 people and you're like, "Okay, these are the guys we need to worry about."
So then straight away, you're going to look at their business. What are they doing? What's their proposition? All you have to do is read someone's website to get a feel for what they're trying to claim. Right? And then you start mapping it and you go, "These guys are claiming this, and they are the market leaders. They really are." And you can look into Google, and you can understand what people are searching.
You can understand if there is any brand terms being used. There is so much free stuff available. If you're building a brand, start with understanding the market, understanding your own business, talk to your customers, do some external searching. You'll be surprised how many third party research that is available on potentially your target customers. It doesn't have to be industry. But if you're targeting family, there's so much information that you can access.
Simon Dell: There's a lot of really easy tools to do that as well. SurveyMonkey, Google Docs… Doing things like that doesn't take a lot of money. Doesn't take a massive amount of time. It's just a question of sitting down and asking some key questions. Make sure the questions are right. You don't just want to have a lot of yes or no questions. You want people to give you real feedback. "What did you think of this?", those really open questions.
When I do those kind of surveys, I always say to people, "Try to mix up." You still got to have a few yes or no questions, closed answers or questions where they can give you numeric value, but then also intersperse that with some of opportunities for them to give you more open feedback.
I'll never forget an old client that did this. They went out and asked all their very loyal customers, people back over and over again. It's a training organization. They've got all of this feedback which was pretty much just blowing smoke up their arse going, "We think you're great. We think you're wonderful. Don't change anything." This isn't great feedback. There was one line, one of them just said something like, "We really like what you're doing, but maybe the colour scheme of your training organization is a bit in your face?"
Because they use purple everywhere. And when you walk into this place, it was just purple everywhere. They kind of went, "Yeah, it feels really cartoony. It feels not serious enough. We want it to be fun but, perhaps we've gone a bit 70's disco fun." Which is the wrong kind of fun. It's tacky kind of fun. And just that colour change alone offered a little bit more. They repainted the place. It offered a little bit more gravitas to the whole place. When you went there, you kind of felt yeah, it didn't feel so comedy.
Mandy Hall: Peaceful?
Simon Dell: Yeah, that's a good one.
Mandy Hall: Comfortable?
Simon Dell: Peaceful is a good one.
Mandy Hall: Yeah, okay.
Simon Dell: Last question. You've seen a lot of brands. You've worked with a lot of brands, but who do you think is killing it out there. What's a great brand for you? You can't say Apple because everybody says Apple.
Mandy Hall: The brand I'm going to say has been in the press a lot lately, but I'm just so impressed by Canva. This is for me. I believe, and everything we do at Chiefs and Indians is all about walking the talk, so making sure our actions speak louder than the words and the document so to speak.
Canva, their products, their services, they've nailed it. They've nailed their customer understanding. I looked at them the other day – and I don't know how long this has been out, but they now do video. You can create video templates. And I whipped one up in half an hour, just because I was curious. Because sometimes when you're working with smaller businesses, where they do just need to put something together relatively quickly, there are some great tools in Canva.
She has just… I mean, yeah, wow. That's one brand that represents the key components of winning, and has done so successfully at a rapid rate. I'm not saying overnight. In terms of business, it's incredible. I think the last valuation was at $8 billion.
Simon Dell: I made a comment on someone's post on their LinkedIn about their valuation.
Mandy Hall: I had this argument with my partner.
Simon Dell: I cannot see how it's possibly valued at that sort of money.
Mandy Hall: We did the math on it.
Simon Dell: I did the math on it!
Mandy Hall: We had this conversation at home and we worked that…
Simon Dell: Was he on my side or your side?
Mandy Hall: Your side.
Simon Dell: Okay, right.
Mandy Hall: Because he's like, "This is…" and I said "No! You don't understand it. It's fantastic. It really taps into our customer need, you know."
Simon Dell: I just went, "Here's the amount of users that they have. How big is their potential market?"
Mandy Hall: But we just went, "Okay, if you look at this – it's really rough and dirty. If you look at the global population, you take 10%… If 10% spent $10."
Simon Dell: Yup. That's plucking numbers.
Mandy Hall: It is plucking but what else can you do? You've got nothing else to go by, but $8 billion. And people in isolation, I get it. You're bored. You want to do something. You want to learn something. And Canva it is.
Simon Dell: I'm thinking about the people, the number of small businesses. That's how I would go back and measure. The number of small business that potentially use this.
Mandy Hall: It's not just small business. My sister, she's trying to get a marketing job at the moment. And I looked at her CV the other day and her cover letter – and she'd done that in Canva. It's not, it's kids, it's school, it's…
Simon Dell: But she wouldn't pay for this. She's still using the free version.
Mandy Hall: Yeah, but you don't understand. If you want a picture, you can pay a dollar. It's not always about the membership, although the membership is, I'm sure, where she's getting the most money. I've been very conservative with $10 per person.
Simon Dell: We shall agree to disagree.
Mandy Hall: I need an idea like that. I want $8 billion. I want to compute with $8 billion.
Simon Dell: I'm on your partner's side. You can tell him that I completely don't understand. What I was going to finish with, because I never get the chance with these things to say brands that I like.
Mandy Hall: Tell me about it.
Simon Dell: What I was going to say to you… I just wanted to talk about the one that has always really impressed me, that has such longevity, is Lego.
Mandy Hall: How good is the Lego movie?
Simon Dell: You've got a 4-year-old? Was it?
Mandy Hall: Three-year-old.
Simon Dell: Yeah, 3-year-old. So, he's going to start getting, in the next couple of years, what my 4-year-old describes as 'big boy Lego', which I end up putting it together and he destroys, but that's a matter entirely. But the fact that they have managed to create this brand that is so relevant to my 4-year-old, as it was relevant to me when I was 9 years old… And it's had some up and downs in there, but really, to me, it hasn't skipped a beat.
Mandy Hall: But I think they evolved. They continue to evolve. These brands… It's really sad, the current market, because we're seeing some strong brands going into administration, which is really sad. But I think brands like Lego continue to revitalize their brand through their products.
Obviously, they've kept strong to their brand history because you don't want to let go of that, for such a brand that has been around in a long time. It's getting the balance of preserving that heritage, but evolving with today's market and kid's needs. They've got a movie. They've sort of tried different things. Do they have computer games?
Simon Dell: Yeah, lots of computer games.
Mandy Hall: They've diversified so that the Lego has remained still strong as a toy, but people have taken Lego throughout their life, probably later, than when they first engaged with it.
Simon Dell: There was the TV show, Lego Masters.
Mandy Hall: And the adults love it. Well, I'm talking about the movie, but I haven't actually watched it. So many people I know, adults like The Simpsons, they love it. And so, they've hit the nail on the head.
Simon Dell: The interesting thing with their business model was when they decided they were going to be more than just bricks, and that they were going to branch out, they originally started to do it all themselves internally. The Lego, the destination venues, the theme parks, the films, the TV – all of these was all done internally, different departments within.
And that put a massive strain on their revenue profitability. I think there was a couple of years in there where they made quite hefty losses. It was perhaps the late 90's to the early 2000's where it was looking quite dodgy for them as a brand, as an organization. And what they realized was they said "No, no, no. Our core business is making building blocks in various different themes."
The Star Wars ones, the Eiffel Tower ones, and all of those kind of things. "That's what we do. Anything else, we still want to do it but we are going to license our stuff out. We are going to let someone else do it." So, people that are good with making films can go and make Lego films. People that are good with TV shows can go and make Lego TV shows.
Mandy Hall: They're just licensing the brand.
Simon Dell: Just licensing the stuff out.
Mandy Hall: Yeah, yeah.
Simon Dell: And we then have a licensing department within the organization. They're the people that control that stuff. But then that licensing department, I assume, is working very, very tightly with the branding within the organization.
Mandy Hall: It's about building the brand.
Simon Dell: And making sure that when the license is executed properly, then everything stays completely to the brand values.
Mandy Hall: Yeah, that's a great one.
Simon Dell: Thank you very much for being on here today. Do you want to do the shameless plug where people can find you and how they can get in contact with you?
Mandy Hall: Well, yep. Sure. You can find me at www.chiefsandindians.com and we are on social media, little light on social media but we are there, on Instagram, LinkedIn and little bit of Facebook.
Simon Dell: Thank you. Thank you very much for your time.
Mandy Hall: Thanks for having me, Simon.
PODCAST EP 75
Marketing to the over 50's with Jon Warner
On Episode 75 of the Paper Planes Podcast Simon chats with Jon Warner, CEO of Silver Moonshots.
Simon chats with Holly Cardew, Founder and CEO of Pixc.
Relationship Marketing with Sophie Musumeci
On Episode 77 of the Paper Planes Podcast, Simon chats with Sophie Musumeci, CEO and Founder of Real Entrepreneur Women.
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